Receiving Helpdesk

what are the requirements for a probability distribution

by Hazel Cassin Published 2 years ago Updated 2 years ago

Three Requirements for probability distribution :

  • The random variable is associated with numerical.
  • The sum of the probabilities has to be equal to 1, discounting any round off error.
  • Each individual probability must be a number between 0 and 1, inclusive. Sets found in the same folder.

Three Requirements for probability distribution :
  • The random variable is associated with numerical.
  • The sum of the probabilities has to be equal to 1, discounting any round off error.
  • Each individual probability must be a number between 0 and 1, inclusive.

Full Answer

How do I create a probability distribution?

where:

  • x_range: The range of numeric x values.
  • prob_range: The range of probabilities associated with each x value.
  • lower_limit: The lower limit on the value for which you want a probability.
  • upper_limit: The upper limit on the value for which you want a probability. Optional.

Which distribution to use for a probability problem?

What is a Probability Distribution?

  • Probability Distribution Prerequisites. To understand probability distributions, it is important to understand variables. ...
  • Probability Distributions. An example will make clear the relationship between random variables and probability distributions. ...
  • Cumulative Probability Distributions. ...
  • Uniform Probability Distribution

Does a probability distribution have to be equal to one?

The sum of all probabilities for all possible values must equal 1. Furthermore, the probability for a particular value or range of values must be between 0 and 1. Probability distributions describe the dispersion of the values of a random variable. Consequently, the kind of variable determines the type of probability distribution.

How to make a probability distribution?

μ = Σx * P (x) where: x: Data value. P (x): Probability of value. For example, consider our probability distribution table for the soccer team: The mean number of goals for the soccer team would be calculated as: μ = 0*0.18 + 1*0.34 + 2*0.35 + 3*0.11 + 4*0.02 = 1.45 goals. 3.

What are the two requirements of a probability distribution?

What are the two requirements for a discrete probability distribution? The first rule states that the sum of the probabilities must equal 1. The second rule states that each probability must be between 0 and 1, inclusive.

What conditions must hold for a probability distribution to be acceptable?

The probability of any event must be positive. So in other words, the probably distribution must not contain a negative value. It should be between zero and 1 because the probability has to be written around one can be negative. The second one, the probability of any event must not exceed one.

What are the properties of a probability distribution?

A probability distribution depicts the expected outcomes of possible values for a given data generating process. Probability distributions come in many shapes with different characteristics, as defined by the mean, standard deviation, skewness, and kurtosis.

Which of the following is not a requirement of a probability distribution?

Which of the following is not a requirement of the binomial probability distribution? The correct answer is B. The trials must be dependent.

How do you know if its a probability distribution or not?

0:011:22Determine if a Table Represents a Probability Distribution - YouTubeYouTubeStart of suggested clipEnd of suggested clipSo looking at the first table the probabilities are under the column p of x or the probability of x.MoreSo looking at the first table the probabilities are under the column p of x or the probability of x. Notice how each of the probabilities are on the closed interval from zero to one.

What must be true for a number to be a probability?

A probability must be between zero and one. (c) Explain why 120% cannot be the probability of some event. A probability must be between zero and one. (d) Can the number 0.56 be the probability of an event?

What is a valid probability distribution?

Step 1: Determine whether each probability is greater than or equal to 0 and less than or equal to 1. Step 2: Determine whether the sum of all of the probabilities equals 1. Step 3: If Steps 1 and 2 are both true, then the probability distribution is valid.

Which of the following conditions is not necessary for a distribution to be a binomial distribution?

As a general rule, the binomial distribution should not be applied to observations from a simple random sample (SRS) unless the population size is at least 10 times larger than the sample size.

Which of the following is not a required of a binomial distribution?

We note that a binomial distribution requires that there are only two possible outcomes (a success or a failure) and thus "three or more outcomes" is not one of the requirements for a binomial distribution.

Which if the following conditions is not necessary for a distribution to be a binomial distribution *?

The probability of success must be the same for all the trials.

What are the two types of probability distributions?

Probability distributions describe the dispersion of the values of a random variable. Consequently, the kind of variable determines the type of probability distribution. For a single random variable, statisticians divide distributions into the following two types: 1 Discrete probability distributions for discrete variables 2 Probability density functions for continuous variables

Why do I like probability distribution plots?

As I mentioned, I really like probability distribution plots because they make distribution properties crystal clear. In the example above, we used the normal distribution. Because that distribution is so well-known, you might have guessed the general appearance of the chart. Now, let’s look at a less intuitive example.

What is a discrete probability function?

Discrete probability functions are also known as probability mass functions and can assume a discrete number of values. For example, coin tosses and counts of events are discrete functions. These are discrete distributions because there are no in-between values.

What statistic do you use for a hypothesis test?

For example, t-tests use t-values, ANOVA uses F-values, and Chi-square tests use chi-square values. Hypothesis tests use the probability distributions of these test statistics to calculate p-values. That’s right, p-values come from these distributions!

What is the importance of inferential statistics?

A vital concept in inferential statistics is that the particular random sample that you draw for a study is just one of a large number of possible samples that you could have pulled from your population of interest. Understanding this broader context of all possible samples and how your study’s sample fits within it provides valuable information.

image
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 1 2 3 4 5 6 7 8 9
8.3.21PHP Version1.2sRequest Duration2MBMemory UsageGET {post}Route
  • warninglog[05:19:11] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\QueryFormatter:...
  • warninglog[05:19:11] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\QueryFormatter:...
  • warninglog[05:19:11] LOG.warning: Callables of the form ["Swift_SmtpTransport", "Swift_Transport_EsmtpTranspor...
  • warninglog[05:19:11] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\SimpleFormatter...
  • warninglog[05:19:11] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\SimpleFormatter...
  • warninglog[05:19:11] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[05:19:11] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[05:19:11] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[05:19:11] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • Booting (11.82ms)
  • Application (1.19s)
  • 1 x Application (98.98%)
    1.19s
    1 x Booting (0.99%)
    11.82ms
    7 templates were rendered
    • themes.DevBlog.content.post (resources/views/themes/DevBlog/content/post.blade.php)34blade
      Params
      0
      post
      1
      postContent
      2
      author
      3
      updated_at
      4
      bing_rich_snippet_text
      5
      bing_rich_snippet_link
      6
      bing_related_keywords
      7
      google_related_keywords
      8
      bing_news_title
      9
      bing_news_description
      10
      bing_videos
      11
      bing_images
      12
      bing_search_result_title
      13
      bing_search_result_description
      14
      bing_search_result_url
      15
      bing_paa_questions
      16
      bing_paa_answers
      17
      bing_slider_faq_questions
      18
      bing_slider_faq_answers
      19
      bing_pop_faq_questions
      20
      bing_pop_faq_answers
      21
      bing_tab_faq_questions
      22
      bing_tab_faq_answers
      23
      google_faq_questions
      24
      google_faq_answers
      25
      google_rich_snippet
      26
      google_search_result
      27
      indexedArray
      28
      total_images
      29
      total_videos
      30
      settings
      31
      url_current
      32
      menus
      33
      sidebar
    • themes.DevBlog.layouts.master (resources/views/themes/DevBlog/layouts/master.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.head (resources/views/themes/DevBlog/panels/head.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.header (resources/views/themes/DevBlog/panels/header.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.navbar (resources/views/themes/DevBlog/panels/navbar.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.footer (resources/views/themes/DevBlog/panels/footer.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.scripts (resources/views/themes/DevBlog/panels/scripts.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    uri
    GET {post}
    middleware
    web, checkdate
    as
    post.show
    controller
    App\Http\Controllers\Frontend\json_data\PostController@show
    namespace
    where
    file
    app/Http/Controllers/Frontend/json_data/PostController.php:18-166
    7 statements were executed1.17s
    • select * from `posts` where `published_at` <= '2025-06-09 05:19:11' and `slug` = 'what-are-the-requirements-for-a-probability-distribution' and `posts`.`deleted_at` is null limit 1
      3.35ms/app/Providers/RouteServiceProvider.php:54receivinghelpdeskask
      Metadata
      Bindings
      • 0. 2025-06-09 05:19:11
      • 1. what-are-the-requirements-for-a-probability-distribution
      Backtrace
      • 15. /app/Providers/RouteServiceProvider.php:54
      • 18. /vendor/laravel/framework/src/Illuminate/Routing/Router.php:842
      • 19. Route binding:39
      • 20. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 21. /vendor/laravel/framework/src/Illuminate/Foundation/Http/Middleware/VerifyCsrfToken.php:78
    • select * from `json_post_contents` where `json_post_contents`.`post_id` = 217065 and `json_post_contents`.`post_id` is not null and `rewrite_id` = 0
      8.76msmiddleware::checkdate:30receivinghelpdeskask
      Metadata
      Bindings
      • 0. 217065
      • 1. 0
      Backtrace
      • 19. middleware::checkdate:30
      • 20. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 21. /vendor/laravel/jetstream/src/Http/Middleware/ShareInertiaData.php:61
      • 22. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 23. /vendor/laravel/framework/src/Illuminate/Routing/Middleware/SubstituteBindings.php:50
    • select * from `nova_menu_menus` where `slug` = 'header' limit 1
      270μs/vendor/outl1ne/nova-menu-builder/src/helpers.php:32receivinghelpdeskask
      Metadata
      Bindings
      • 0. header
      Backtrace
      • 15. /vendor/outl1ne/nova-menu-builder/src/helpers.php:32
      • 17. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 18. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 19. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
      • 20. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
    • select * from `nova_menu_menu_items` where `nova_menu_menu_items`.`menu_id` = 1 and `nova_menu_menu_items`.`menu_id` is not null and `parent_id` is null order by `parent_id` asc, `order` asc, `name` asc
      240μs/vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35receivinghelpdeskask
      Metadata
      Bindings
      • 0. 1
      Backtrace
      • 19. /vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35
      • 20. /vendor/outl1ne/nova-menu-builder/src/helpers.php:33
      • 22. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 23. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 24. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
    • select * from `nova_menu_menu_items` where `nova_menu_menu_items`.`parent_id` in (1) order by `order` asc
      240μs/vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35receivinghelpdeskask
      Metadata
      Backtrace
      • 24. /vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35
      • 25. /vendor/outl1ne/nova-menu-builder/src/helpers.php:33
      • 27. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 28. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 29. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
    • select `id`, `post_title`, `slug` from `posts` where `status` = 'publish' and `posts`.`deleted_at` is null order by RAND() limit 10
      1.15s/app/View/Composers/SidebarView.php:22receivinghelpdeskask
      Metadata
      Bindings
      • 0. publish
      Backtrace
      • 14. /app/View/Composers/SidebarView.php:22
      • 15. /app/View/Composers/SidebarView.php:12
      • 16. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:124
      • 17. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:162
      • 20. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:177
    • select * from `fake_users` where `fake_users`.`id` = 14155 limit 1
      1.22msview::2dd102cf0462e89a4d4d8bc77355d767652bf9aa:15receivinghelpdeskask
      Metadata
      Bindings
      • 0. 14155
      Backtrace
      • 21. view::2dd102cf0462e89a4d4d8bc77355d767652bf9aa:15
      • 23. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:108
      • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 25. /vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
      • 26. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:61
    App\Models\FakeUser
    1
    Outl1ne\MenuBuilder\Models\MenuItem
    1
    Outl1ne\MenuBuilder\Models\Menu
    1
    App\Models\JsonPostContent
    1
    App\Models\Post
    11
        _token
        6LJYVPU8HcZ5tqRm00NGUYPIVNI1ZHJRhILGTdK1
        _previous
        array:1 [ "url" => "https://receivinghelpdesk.com/ask/what-are-the-requirements-for-a-probability-...
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /what-are-the-requirements-for-a-probability-distribution
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:25 [ "cookie" => array:1 [ 0 => "_pk_id.64.7c30=51a060e15b8e7324.1749426544.; _pk_ses.64.7c30=1; XSRF-TOKEN=eyJpdiI6IkF2ZzBJK0ptbjhWNWN3OGJnRWNRbEE9PSIsInZhbHVlIjoiUmEzcDhwTFBvN21qcS9tK0lMQTMvcHFkQm9ZNGpBS1lmL3VGN3ZNd0pNQ1BOSTNIdlRFVDRJREFIKzhPc0h3bGlqZC9uNWRZQ0pxaVJjcHhQTEd1S3BJWXF4SXJVMzZLd0c3RmgrYUJKRHFuRTF2dm9ZMEc2L0JNc0ozVW9wMUciLCJtYWMiOiI0OTQzZDgwMjRlNTg1ZWE4MDI0NDRiNDM3M2UzZjNjYmY2MGNjYWZlYmU4MDBmN2FmZjdjYzMwZTQxNjYzNDYxIiwidGFnIjoiIn0%3D; askhelpdesk_session=eyJpdiI6IkhXVVplSnhwU0xTU09ZTURmTnozMUE9PSIsInZhbHVlIjoiVmZ3dEJ4YTlLUFJheXhPdW9zWEk0bFBMSXh0WlFBVWQ5MllJaW5QdVVwdlJCOGtKSjI3cmpLV2k1dzJ1ZEs3ZkdTMlZ0SThkT0tReDhOa0tVQWVIVjRnVldua3pzR21VZG8xY28wK2ZRQTB2dFdRdTFyQi9mWVJnczlrRnZvaEQiLCJtYWMiOiIwYmU2MWRhMmI1OTg3OGE2YzZiMmIzY2YzYWNhYzMzZDE4NzM4MmUyZDQzZTU1Y2JlMWY1NWUyY2VkMzUyMzFhIiwidGFnIjoiIn0%3D_pk_id.64.7c30=51a060e15b8e7324.1749426544.; _pk_ses.64.7c30=1; XSRF-TOKEN=eyJpdiI6IkF2ZzBJK0ptbjhWNWN3OGJnRWNRbEE9PSIsInZhbHVlIjoiUmEzcDhwTFBvN21qcS9tK0lMQTMvc" ] "cf-ipcountry" => array:1 [ 0 => "US" ] "cf-connecting-ip" => array:1 [ 0 => "216.73.216.86" ] "cdn-loop" => array:1 [ 0 => "cloudflare; loops=1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "accept-encoding" => array:1 [ 0 => "gzip, br" ] "cf-ray" => array:1 [ 0 => "94cc48c9beeeeae2-ORD" ] "priority" => array:1 [ 0 => "u=0, i" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "cf-visitor" => array:1 [ 0 => "{"scheme":"https"}" ] "connection" => array:1 [ 0 => "close" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "216.73.216.86, 172.70.126.217" ] "x-server-addr" => array:1 [ 0 => "154.12.239.204" ] "host" => array:1 [ 0 => "receivinghelpdesk.com" ] ]
        request_server
        0 of 0
        array:56 [ "USER" => "runcloud" "HOME" => "/home/runcloud" "SCRIPT_NAME" => "/ask/index.php" "REQUEST_URI" => "/ask/what-are-the-requirements-for-a-probability-distribution" "QUERY_STRING" => "" "REQUEST_METHOD" => "GET" "SERVER_PROTOCOL" => "HTTP/1.0" "GATEWAY_INTERFACE" => "CGI/1.1" "REDIRECT_URL" => "/ask/what-are-the-requirements-for-a-probability-distribution" "REMOTE_PORT" => "48442" "SCRIPT_FILENAME" => "/home/runcloud/webapps/ReceivingHelpDesk/ask/index.php" "SERVER_ADMIN" => "you@example.com" "CONTEXT_DOCUMENT_ROOT" => "/home/runcloud/webapps/ReceivingHelpDesk/" "CONTEXT_PREFIX" => "" "REQUEST_SCHEME" => "http" "DOCUMENT_ROOT" => "/home/runcloud/webapps/ReceivingHelpDesk/" "REMOTE_ADDR" => "172.70.126.217" "SERVER_PORT" => "80" "SERVER_ADDR" => "127.0.0.1" "SERVER_NAME" => "receivinghelpdesk.com" "SERVER_SOFTWARE" => "Apache/2.4.63 (Unix) OpenSSL/1.1.1f" "SERVER_SIGNATURE" => "" "LD_LIBRARY_PATH" => "/RunCloud/Packages/apache2-rc/lib" "PATH" => "/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin" "HTTP_COOKIE" => "_pk_id.64.7c30=51a060e15b8e7324.1749426544.; _pk_ses.64.7c30=1; XSRF-TOKEN=eyJpdiI6IkF2ZzBJK0ptbjhWNWN3OGJnRWNRbEE9PSIsInZhbHVlIjoiUmEzcDhwTFBvN21qcS9tK0lMQTMvcHFkQm9ZNGpBS1lmL3VGN3ZNd0pNQ1BOSTNIdlRFVDRJREFIKzhPc0h3bGlqZC9uNWRZQ0pxaVJjcHhQTEd1S3BJWXF4SXJVMzZLd0c3RmgrYUJKRHFuRTF2dm9ZMEc2L0JNc0ozVW9wMUciLCJtYWMiOiI0OTQzZDgwMjRlNTg1ZWE4MDI0NDRiNDM3M2UzZjNjYmY2MGNjYWZlYmU4MDBmN2FmZjdjYzMwZTQxNjYzNDYxIiwidGFnIjoiIn0%3D; askhelpdesk_session=eyJpdiI6IkhXVVplSnhwU0xTU09ZTURmTnozMUE9PSIsInZhbHVlIjoiVmZ3dEJ4YTlLUFJheXhPdW9zWEk0bFBMSXh0WlFBVWQ5MllJaW5QdVVwdlJCOGtKSjI3cmpLV2k1dzJ1ZEs3ZkdTMlZ0SThkT0tReDhOa0tVQWVIVjRnVldua3pzR21VZG8xY28wK2ZRQTB2dFdRdTFyQi9mWVJnczlrRnZvaEQiLCJtYWMiOiIwYmU2MWRhMmI1OTg3OGE2YzZiMmIzY2YzYWNhYzMzZDE4NzM4MmUyZDQzZTU1Y2JlMWY1NWUyY2VkMzUyMzFhIiwidGFnIjoiIn0%3D_pk_id.64.7c30=51a060e15b8e7324.1749426544.; _pk_ses.64.7c30=1; XSRF-TOKEN=eyJpdiI6IkF2ZzBJK0ptbjhWNWN3OGJnRWNRbEE9PSIsInZhbHVlIjoiUmEzcDhwTFBvN21qcS9tK0lMQTMvc" "HTTP_CF_IPCOUNTRY" => "US" "HTTP_CF_CONNECTING_IP" => "216.73.216.86" "HTTP_CDN_LOOP" => "cloudflare; loops=1" "HTTP_SEC_FETCH_MODE" => "navigate" "HTTP_SEC_FETCH_SITE" => "none" "HTTP_ACCEPT" => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" "HTTP_USER_AGENT" => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" "HTTP_UPGRADE_INSECURE_REQUESTS" => "1" "HTTP_SEC_CH_UA_PLATFORM" => ""Windows"" "HTTP_SEC_CH_UA_MOBILE" => "?0" "HTTP_SEC_CH_UA" => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99"" "HTTP_CACHE_CONTROL" => "no-cache" "HTTP_PRAGMA" => "no-cache" "HTTP_ACCEPT_ENCODING" => "gzip, br" "HTTP_CF_RAY" => "94cc48c9beeeeae2-ORD" "HTTP_PRIORITY" => "u=0, i" "HTTP_SEC_FETCH_DEST" => "document" "HTTP_SEC_FETCH_USER" => "?1" "HTTP_CF_VISITOR" => "{"scheme":"https"}" "HTTP_CONNECTION" => "close" "HTTP_X_FORWARDED_PROTO" => "https" "HTTP_X_FORWARDED_FOR" => "216.73.216.86, 172.70.126.217" "HTTP_X_SERVER_ADDR" => "154.12.239.204" "HTTP_HOST" => "receivinghelpdesk.com" "HTTPS" => "on" "REDIRECT_STATUS" => "200" "REDIRECT_HTTPS" => "on" "FCGI_ROLE" => "RESPONDER" "PHP_SELF" => "/ask/index.php" "REQUEST_TIME_FLOAT" => 1749426551.3345 "REQUEST_TIME" => 1749426551 ]
        request_cookies
        0 of 0
        array:4 [ "_pk_id_64_7c30" => null "_pk_ses_64_7c30" => null "XSRF-TOKEN" => "6LJYVPU8HcZ5tqRm00NGUYPIVNI1ZHJRhILGTdK1" "askhelpdesk_session" => "UN6eBLovp4RQevMq79J1i1IPqLCouPYiPo6rJOyw" ]
        response_headers
        0 of 0
        array:7 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "private, must-revalidate" ] "date" => array:1 [ 0 => "Sun, 08 Jun 2025 23:49:11 GMT" ] "pragma" => array:1 [ 0 => "no-cache" ] "expires" => array:1 [ 0 => -1 ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IktJM3Fsay9lL0N5RjIvdUFRc3pmaXc9PSIsInZhbHVlIjoiVE5yN3pIRjNpaERMZ2VXNUpjZmhlblNjYkc0OUtCcEpnMjErNDMzL1RDVzg5WjZNV1lycDNCalRGK2MwZFZxWVBGa0JVbnl2SEpjdENPNzFqOEdVUitHNDZuU2FnSnpjQnJvVTZJOURoOFFvWmV5ZU5mejB1Zkxla1d6a09WMVkiLCJtYWMiOiJmNGU0NjA1NTA3MmE5YTM0ZmUwYzA4M2JlZmU0NmM1N2UwZWM4M2JkYjBmNTc3NjQ5MTJhZjAwODA5M2JlMzE3IiwidGFnIjoiIn0%3D; expires=Mon, 09-Jun-2025 01:49:12 GMT; Max-Age=7200; path=/; samesite=laxXSRF-TOKEN=eyJpdiI6IktJM3Fsay9lL0N5RjIvdUFRc3pmaXc9PSIsInZhbHVlIjoiVE5yN3pIRjNpaERMZ2VXNUpjZmhlblNjYkc0OUtCcEpnMjErNDMzL1RDVzg5WjZNV1lycDNCalRGK2MwZFZxWVBGa0JVb" 1 => "askhelpdesk_session=eyJpdiI6IkNaMDE0NXREZFBNSHZSS0Q4MzYzR0E9PSIsInZhbHVlIjoia2JZWXFGWEtKSGNCSG9HNEtoU2g1OThHWWRjL0tIUHR5aE5TdDlrTWRXMEk0SjZZeU55Tm9qOFNoZEFaY25nQmtqQkJMaEFNcWFENERvbkFxenBYd0hQZFQ2ZFFOQXo1eEJGeVNQUXNESWtxWXUya1ZBcUIzb3h0S29pSk9ib08iLCJtYWMiOiI2YzNiZTNjNTgzNDgxNjJlMGNjMGIyZTUzMmM5MTZjODkxNGUyMTRmM2E1N2QwMDk1ZmQ5MjhkZDM0NTZmNDhjIiwidGFnIjoiIn0%3D; expires=Mon, 09-Jun-2025 01:49:12 GMT; Max-Age=7200; path=/; httponly; samesite=laxaskhelpdesk_session=eyJpdiI6IkNaMDE0NXREZFBNSHZSS0Q4MzYzR0E9PSIsInZhbHVlIjoia2JZWXFGWEtKSGNCSG9HNEtoU2g1OThHWWRjL0tIUHR5aE5TdDlrTWRXMEk0SjZZeU55Tm9qOFNoZEFaY25n" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IktJM3Fsay9lL0N5RjIvdUFRc3pmaXc9PSIsInZhbHVlIjoiVE5yN3pIRjNpaERMZ2VXNUpjZmhlblNjYkc0OUtCcEpnMjErNDMzL1RDVzg5WjZNV1lycDNCalRGK2MwZFZxWVBGa0JVbnl2SEpjdENPNzFqOEdVUitHNDZuU2FnSnpjQnJvVTZJOURoOFFvWmV5ZU5mejB1Zkxla1d6a09WMVkiLCJtYWMiOiJmNGU0NjA1NTA3MmE5YTM0ZmUwYzA4M2JlZmU0NmM1N2UwZWM4M2JkYjBmNTc3NjQ5MTJhZjAwODA5M2JlMzE3IiwidGFnIjoiIn0%3D; expires=Mon, 09-Jun-2025 01:49:12 GMT; path=/XSRF-TOKEN=eyJpdiI6IktJM3Fsay9lL0N5RjIvdUFRc3pmaXc9PSIsInZhbHVlIjoiVE5yN3pIRjNpaERMZ2VXNUpjZmhlblNjYkc0OUtCcEpnMjErNDMzL1RDVzg5WjZNV1lycDNCalRGK2MwZFZxWVBGa0JVb" 1 => "askhelpdesk_session=eyJpdiI6IkNaMDE0NXREZFBNSHZSS0Q4MzYzR0E9PSIsInZhbHVlIjoia2JZWXFGWEtKSGNCSG9HNEtoU2g1OThHWWRjL0tIUHR5aE5TdDlrTWRXMEk0SjZZeU55Tm9qOFNoZEFaY25nQmtqQkJMaEFNcWFENERvbkFxenBYd0hQZFQ2ZFFOQXo1eEJGeVNQUXNESWtxWXUya1ZBcUIzb3h0S29pSk9ib08iLCJtYWMiOiI2YzNiZTNjNTgzNDgxNjJlMGNjMGIyZTUzMmM5MTZjODkxNGUyMTRmM2E1N2QwMDk1ZmQ5MjhkZDM0NTZmNDhjIiwidGFnIjoiIn0%3D; expires=Mon, 09-Jun-2025 01:49:12 GMT; path=/; httponlyaskhelpdesk_session=eyJpdiI6IkNaMDE0NXREZFBNSHZSS0Q4MzYzR0E9PSIsInZhbHVlIjoia2JZWXFGWEtKSGNCSG9HNEtoU2g1OThHWWRjL0tIUHR5aE5TdDlrTWRXMEk0SjZZeU55Tm9qOFNoZEFaY25n" ] ]
        session_attributes
        0 of 0
        array:4 [ "_token" => "6LJYVPU8HcZ5tqRm00NGUYPIVNI1ZHJRhILGTdK1" "_previous" => array:1 [ "url" => "https://receivinghelpdesk.com/ask/what-are-the-requirements-for-a-probability-distribution" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]