Discrete Random Variable: Difference between revisions
(Created page with "Category:Statistics A random variable is '''discrete''' if the values it can take on within an interval is ''finite''. = PMF and CDF = The '''probability mass function (PMF)''' describes the probability distribution over a discrete random variable. <math>p(x) = P(X = x)</math> The '''cumulative distribution function (CDF)''' specifies the probability of an observation being equal to or less than a given value. <math>F(x) = P(X \leq x)</math> We usually have tabl...") |
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[[Category:Statistics]] | [[Category:Statistics]] | ||
[[Category:Distribution (Statistics)]] | |||
A random variable is '''discrete''' if the values it can take on within an interval is ''finite''. | A random variable is '''discrete''' if the values it can take on within an interval is ''finite''. | ||
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We usually have tables for these in the case of discrete random variables. | We usually have tables for these in the case of discrete random variables. | ||
= Statistics = | |||
Expected value (mean): | |||
<math> | |||
\mu = E(X) = \sum x_i P(X = x_i) | |||
</math> | |||
= Distributions = | |||
== Bernoulli == | |||
The '''bernoulli distribution''' describes the random variable of an experiment that has two outcomes and is performed once. The outcomes are either ''success'' or ''failure''. | |||
<math> | |||
X \sim Bernoulli(p) | |||
</math> | |||
=== PMF === | |||
<math> | |||
p(1) = p, p(0) = 1 - p | |||
</math> | |||
=== Statistics === | |||
<math> | |||
\mu = p | |||
</math> | |||
<math> | |||
\sigma^2_X = p (1 - p) | |||
</math> | |||
== Binomial == | |||
Repeating a bernoulli experiment <math>b</math> times and we get a '''binomial random variable'''. | |||
Consider an experiment with exactly two possible outcomes, conducted n times independently. | |||
<math> | |||
X \sim Binomial(n, p) | |||
</math> | |||
I'm sleep I'll write the details later. It should be on the equation sheet. | |||
Revision as of 07:32, 19 March 2024
A random variable is discrete if the values it can take on within an interval is finite.
PMF and CDF
The probability mass function (PMF) describes the probability distribution over a discrete random variable.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = P(X = x)}
The cumulative distribution function (CDF) specifies the probability of an observation being equal to or less than a given value.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x) = P(X \leq x)}
We usually have tables for these in the case of discrete random variables.
Statistics
Expected value (mean):
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = E(X) = \sum x_i P(X = x_i) }
Distributions
Bernoulli
The bernoulli distribution describes the random variable of an experiment that has two outcomes and is performed once. The outcomes are either success or failure.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim Bernoulli(p) }
PMF
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(1) = p, p(0) = 1 - p }
Statistics
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = p }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2_X = p (1 - p) }
Binomial
Repeating a bernoulli experiment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} times and we get a binomial random variable.
Consider an experiment with exactly two possible outcomes, conducted n times independently.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim Binomial(n, p) }
I'm sleep I'll write the details later. It should be on the equation sheet.
