The last session is review session
A random variable is a function that associates with number with each outcome of the sample space of a random experiment.
A discrete random variable is a random variable whose sample space has a finite or at most a countably infinite number of values
Example: Toss a fair coin and let X be the number of tosses until the third head occurs.
\[ S_X=\{3,4,5,6...\} infinite,ountable \]
A continuous random variable can take any value within a finite or infinite interval of the real number line (-\(\infty\),\(\infty\))*
The cumulative distribution function (CDF) of a random variable X is the function \[ F(x)=P(X\leq x) \]
for \(x \in [-\infty,\infty]\) (in real number line).
It is non-decreasing: If \(a\leq b\), then \(F(a)\leq F(b)\), because \([X\leq a]\subseteq[X\leq b]\).
\(F(-\infty)=0\) and $F()=1 $.
If \(a< b\), then \(P(a<X\leq b)=F(b)-F(a)\).
Example: Toss a fair coin 3 times and let X be the number of heads:
Outcomes: {HHH,HHT,…TTT} |S|=8,
The PMF for x
x | 0 1 2 3
P(x)|(1/8)(3/8)(3/8)(1/8)
The CDF:
For \(x<0, F(x)=P(X\leq x)=0\);
For \(0\leq x<1, F(x)=P(X\leq x)=P(X=0)=1/8\);
For \(1\leq x<2, F(x)=P(X\leq x)=P(X=0)+P(X=1)=1/2\);
For \(2\leq x<3, F(x)=P(X\leq x)=P(X=0)+P(X=1)+P(X=2)=7/8\);
For \(3\leq x, F(x)=P(X\leq x)=P(X=0)+P(X=1)+P(X=2)+P(X=3)=1\);
Let \(x_1<x_2<x_3...\) denote the possible values of X. Then
F is a step function with jumps occurring only at the values of x of \(S_X\). The size of the jump at each x of \(S_X\) equals p(x).
The CDF can be obtained from the PMF:
\[ F(x)=\sum_{x_i\leq x}p(x_i) \]
\[ p(x_i)=F(x_i)-F(x_i-1) for i=2,3,... \]
\[ P(a < X \leq b)=F(b)-F(a)=\sum_{a<x_i<b}p(x_i) \]
Example: Suppose the CDF of a random variable X is given by \[ F(x)=\begin{cases}0,x<-1\\1/3,-1\leq x<1\\1/2,1\leq x<2\\1,2\leq x \end{cases} \]
The places for jump (-1,1,2) is the change of P(x)
For a continuous random variable P(X=x)=0 for all x.
The CDF is a continuous function.
Example: We say X is the uniform in [0,1] random variable if X has CDF
\[ F(x)=\begin{cases}0,x<0\\x,0\leq x<1\\1,1\leq x\end{cases} \]
The probability density function (PDF of a continuous random variable X is a non-negative function f such that
$P(a<X<b) $ = area under f between a and b = \(\int_a^b{f(x)dx}\)
You can also write it with equal sign because P(X=a)=P(X=b)=0
Let X be a continuous random variable with PDF f(x) and CDF F(x). Then
\(\int_{-\infty}^{\infty}{f(x)dx}=1\)
The CDF can be obtained form the PDF:
\[ F(x)=P(X\leq x)=\int^x_{-\infty}f(y)dy \]
\[ f(x)={d\over{dx}}F(x) \]