1. Math 4030-3a Discrete Random Variables

2. Random variable
A function that assigns a numerical value to each possible outcome in the sample space.
One value for one outcome. i.e. different value must mean different outcomes.
However, different outcomes may have the same value.
Random variable may take discrete or continuous values.
Sample space S R
5/28/2018

3. Probability Distribution of a discrete random variable:
A list of probability values corresponding to all values of a discrete random variable X. i.e.
for any value x that the random variable X takes.
5/28/2018

4. Probability histogram and bar chart;
Cumulative distribution function F(x):
If X takes values
then
5/28/2018

5. Mean of a random variable
Mean may or may not exist;
Even exists, may or may not be a finite number;
If X has only finitely many values, then mean exists and is a finite value.
Average value
Balance point
Expect value
5/28/2018

6. Properties of Means:
Linearity:
Warning:
5/28/2018

7. Variance of a random variable
Variance is a “mean”;
Always nonnegative;
Variance may or may not exist;
Even exists, may or may not be a finite number;
If X has only finitely many values, then variance exists and is a finite value.
Measure the variability of X
Spread of the histogram
Un-predictable, risky, …
5/28/2018

8. Properties of variance
Warning:
5/28/2018

9. Chebyshev’s Theorem
If X is a random variable with mean  and standard deviation , then for any number k > 1,
Equivalently,
5/28/2018