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
- Fixed \(n\)
- Each trial has 2 outcomes
- The probability of success is a constant \(p\)
- 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 \(X\):
- \(F(-\infty)\equiv \lim_{y\rightarrow -\infty}F(y)=0\)
- \(F(\infty)\equiv \lim_{y\rightarrow \infty}F(y)=1\)
- \(F(x)\) is a nondecreaseing function
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:
- \(f(x)\geq 0\)
- \(\int^\infty_{-\infty}f(x)dx=1\)
- \(P(a\leq X\leq b) = P(a<X<b)=\int^b_af(x)dx\)
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
- \(E(c)=c\), where \(c\) is constant
- \(E\{cg(X)\}=cE\{g(X)\}\)
- \(E\{g_1(X)+g_2(X)+\cdots+g_n(X)\}=E\{g_1(X)\}+E\{g_2(X)\}+\cdots+E\{g_n(X)\}\)
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
