Week 2
This week we will review basic concepts related to distribution functions and random variables
Learning Outcomes
Monday
Wednesday
Define Moment Generating Functions
Discuss Properties
Important Concepts
Monday
Discrete 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
\[ E(Y)=\sum_y yP(y) \]
Continuous 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
\[ E\{g(X)\}=\int g(x)f(x)dx \] Special properties of the expected value:
- \(E(c)=c\), where \(c\) is constant
- \(E\{cg(X)\}=cE\{g(X)\}\)
- \(E\{g_1(X)+g_2(X)+\cdots+g_n(X)\}=E\{g_1(X)\}+E\{g_2(X)\}+\cdots+E\{g_n(X)\}\)
Wednesday
Moments
The \(k\)th moment is defined as the expectation of the random variable, raised to the \(k\)th power, defined as \(E(X^k)\).
Moment Generating Functions
The moment generating functions is used to obtain the \(k\)th moment. The mgf is defined as
\[ m(c) = E(e^{tX}) \]
The \(k\)th moment can be obtained by taking the \(k\)th derivative of the mgf, with respect to \(c\), and setting \(c\) equal to 0:
\[ E(X^k)=\frac{d^km(c)}{dc}\Bigg|_{c=0} \]
Resources
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| Lecture | Slides | Videos |
|---|---|---|
| Monday | ||
| Wednesday |