Review Random Variables and Distribution Functions

Learning Outcomes

• Review Random Variables

• Review Discrete Random Variables

• Review Continuous Random Variables

Random Variables

A random variable is function that maps the sample space to real value.

Discrete Random Variables

Discrete Random Variables

A random variable is considered to be discrete if it can only map to a finite or countably infinite number of distinct values.

PMF

The probability mass function of discrete variable can be represented by a formula, table, or a graph. The Probability of a random variable Y can be expressed as \(P(Y=y)\) for all values of \(y\).

CDF

The cumulative distribution function provides the \(P(Y\leq y)\) for a random variable \(Y\).

Expected Value

The expected value is the value we expect when we randomly sample from population that follows a specific distribution. The expected value of Y is

\[ E(Y)=\sum_y yP(y) \]

Variance

The variance is the expected squared difference between the random variable and expected value.

\[ Var(Y)=\sum_y\{y-E(Y)\}^2P(y) \]

\[ Var(Y) = E(X^2) - E(X)^2 \]

Known Distributions

Distribution	Parameter(s)	PMF \(P(Y=y)\)
Bernoulli	\(p\)	\(p\)
Binomial	\(n\) and \(p\)	\((^n_y)p^y(1-p)^{n-p}\)
Geometric	\(p\)	\((1-p)^{y-1}p\)
Negative Binomial	\(r\) and \(p\)	\((^{y-1}_{r-1})p^{r-1}(1-p)^{y-r}\)
Hypergeometric	\(N\), \(n\), and \(r\)	\(\frac{(^r_y)(^{N-r}_{n-y})}{(^N_n)}\)
Poisson	\(\lambda\)	\(\frac{\lambda^y}{y!} e^{-\lambda}\)

Binomial Distribution

Binomial Distribution

An experiment is said to follow a binomial distribution if

1. Fixed \(n\)
2. Each trial has 2 outcomes
3. The probability of success is a constant \(p\)
4. The trials are independent of each

\(P(X=x)=(^n_x)p^x(1-p)^{n-x}\)

Expected Value of a Binomial Distribution

Continued

Poisson Distribution

Poisson Distribution

The poisson distribution describes an experiment that measures that occurrence of an event at specific point and/or time period.

\(P(X=x)=\frac{\lambda^x}{x!}e^{-\lambda}\)

Expected Value of a Poisson Distribution

Continuous Random Variables

Continuous Random Variables

A random variable \(X\) is considered continuous if the \(P(X=x)\) does not exist.

CDF

The cumulative distribution function of \(X\) provides the \(P(X\leq x)\), denoted by \(F(x)\), for the domain of \(X\).

Properties of the CDF of \(X\):

1. \(F(-\infty)\equiv \lim_{y\rightarrow -\infty}F(y)=0\)
2. \(F(\infty)\equiv \lim_{y\rightarrow \infty}F(y)=1\)
3. \(F(x)\) is a nondecreaseing function

PDF

The probability density function of the random variable \(X\) is given by

\[ f(x)=\frac{dF(x)}{d(x)}=F^\prime(x) \]

wherever the derivative exists.

Properties of pdfs:

1. \(f(x)\geq 0\)
2. \(\int^\infty_{-\infty}f(x)dx=1\)
3. \(P(a\leq X\leq b) = P(a<X<b)=\int^b_af(x)dx\)

Expected Value

The expected value for a continuous distribution is defined as

\[ E(X)=\int x f(x)dx \]

The expectation of a function \(g(X)\) is defined as

\[ E\{g(X)\}=\int g(x)f(x)dx \]

Expected Value Properties

1. \(E(c)=c\), where \(c\) is constant
2. \(E\{cg(X)\}=cE\{g(X)\}\)
3. \(E\{g_1(X)+g_2(X)+\cdots+g_n(X)\}=E\{g_1(X)\}+E\{g_2(X)\}+\cdots+E\{g_n(X)\}\)

Variance

The variance of continuous variable is defined as

\[ Var(X) = E[\{X-E(X)\}^2] = \int \{X-E(X)\}^2 f(x)dx \]

Uniform Distribution

Uniform Distribution

A random variable is said to follow uniform distribution if the density function is constant between two parameters.

\[ f(x) = \left\{\begin{array}{cc} \frac{1}{b-a} & a \leq x \leq b\\ 0 & \mathrm{elsewhere} \end{array}\right. \]

Expected Value

Normal Distribution

Normal Distribution

A random variable is said to follow a normal distribution if the the frequency of occurrence follow a Gaussian function.

\[ f(x)=\frac{1}{\sqrt{2\pi \sigma^2}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\} \]

Expected Value

Continued