Random Variables#

A real-valued function defined on the outcome of a probability experiment is called a random variable.

If \(X\) is a random variable, then the function \(F(x)\) defined by

\[ F(x)=P\{X \leq x\} \]

is called the distribution function of \(X\). All probabilities concerning \(X\) can be stated in terms of \(F\). A random variable whose set of possible values is either finite or countably infinite is called discrete. If \(X\) is a discrete random variable, then the function

\[ p(x)=P\{X=x\} \]

Expectation#

\(E[X]\) defined by

\[ E[X]=\sum_{x: p(x)>0} x p(x) \]

is called the expected value of \(X . E[X]\) is also commonly called the mean or the expectation of \(X\).

A useful identity states that, for a function \(g\),

\[ E[g(X)]=\sum_{x: p(x)>0} g(x) p(x) \]

Properties of Expectation#

The expected value of a sum of random variables is equal to the sum of their expected values. That is,

\[ E\left[\sum_{i=1}^{n} X_{i}\right]=\sum_{i=1}^{n} E\left[X_{i}\right] \]

Variance#

Variance of a random variable \(X\), denoted by \(\operatorname{Var}(X)\), is defined by

\[ \operatorname{Var}(X)=E\left[(X-E[X])^{2}\right] \]

The variance, which is equal to the expected square of the difference between \(X\) and its expected value, is a measure of the spread of the possible values of \(X\). A useful identity is

\[ \operatorname{Var}(X)=E\left[X^{2}\right]-(E[X])^{2} \]

Standard Deviation (\(\sigma\))#

The quantity \(\sqrt{\operatorname{Var}(X)}\) is called the standard deviation of \(X\).

Binomial (Binom(\(n\), \(p\)))#

The random variable X whose probability mass function is given by

\[\begin{split} p(i)=\left(\begin{array}{c} n \\ i \end{array}\right) p^{i}(1-p)^{n-i} \quad i=0, \ldots, n \end{split}\]

is said to be a binomial random variable with parameters \(n\) and \(p\). Such a random variable can be interpreted as being the number of successes that occur when \(n\) independent trials, each of which results in a success with probability \(p\), are performed. Its mean and variance are given by

\[ E[X]=n p \quad \operatorname{Var}(X)=n p(1-p) \]

Poisson#

The random variable \(X\) whose probability mass function is given by

\[ p(i)=\frac{e^{-\lambda} \lambda^{i}}{i !} \quad i \geq 0 \]

is said to be a Poisson random variable with parameter \(\lambda\). If a large number of (approximately) independent trials are performed, each having a small probability of being successful, then the number of successful trials that result will have a distribution which is approximately that of a Poisson random variable.

The mean and variance of a Poisson random variable are both equal to its parameter \(\lambda\). That is,

\[ E[X]=\operatorname{Var}(X)=\lambda \]

Geometric#

The random variable \(X\) whose probability mass function is given by

\[ p(i)=p(1-p)^{i-1} \quad i=1,2, \ldots \]

is said to be a geometric random variable with parameter \(p\). Such a random variable represents the trial number of the first success when each trial is independently a success with probability \(p\). Its mean and variance are given by

\[ E[X]=\frac{1}{p} \quad \operatorname{Var}(X)=\frac{1-p}{p^{2}} \]

Negative binomial#

The random variable \(X\) whose probability mass function is given by

\[\begin{split} p(i)=\left(\begin{array}{l} i-1 \\ r-1 \end{array}\right) p^{r}(1-p)^{i-r} \quad i \geq r \end{split}\]

is said to be a negative binomial random variable with parameters \(r\) and \(p\). Such a random variable represents the trial number of the \(r\) th success when each trial is independently a success with probability \(p\). Its mean and variance are given by

\[ E[X]=\frac{r}{p} \quad \operatorname{Var}(X)=\frac{r(1-p)}{p^{2}} \]

Hypergeometric#

A hypergeometric random variable \(X\) with parameters \(n, N\), and \(m\) represents the number of white balls selected when \(n\) balls are randomly chosen from an urn that contains \(N\) balls of which \(m\) are white. The probability mass function of this random variable is given by

\[\begin{split} p(i)=\frac{\left(\begin{array}{c} m \\ i \end{array}\right)\left(\begin{array}{c} N-m \\ n-i \end{array}\right)}{\left(\begin{array}{c} N \\ n \end{array}\right)} \quad i=0, \ldots, m \end{split}\]

With \(p=m / N\), its mean and variance are

\[ E[X]=n p \quad \operatorname{Var}(X)=\frac{N-n}{N-1} n p(1-p) \]