1. Means and Variances
of Random Variables

2. Means and Variances of Random Variables
The Expected Value of a Random Variable E(X) = 𝜇 𝑥 =
mean of a random variable = 𝑥 𝑖 𝑝 𝑖
Ex. Rolling 2 dice
X- outcomes
P- probability outcome occurs
X
2 3
P
1
2 3 4 5 6 5 4 3
36 36 36 36 36 36 36 36 36
𝑥𝑖 𝑝𝑖
4 5 6 7 8 9 10 11 12
2 1
36 36
2 6 12 20 30 42 40 36 30 22 12
36 36 36 36 36 36 36 36 36 36 36
E(X) = 𝜇 𝑥 =
252
36
=7
Therefore the expected outcome or the mean outcome of rolling 2 dice is 7

3. Variances of Random Variables
Variance of Random Variables: Var(X) = 𝜎 2 = (𝑥 𝑖 − 𝜇 𝑥 )2 ∗ 𝑝 𝑖
Ex. The probability of selling X number of cars is given in the table below. Find the standard
deviation for selling X cars.
Number of Cars (X)
Probability of X
0
0.3
1
0.4
2
0.2
3
0.1
Calculate 𝜇 𝑥
𝑥𝑖 𝑝𝑖 0
0.4
0.4
0.3
(𝑥 𝑖 − 𝜇)
-1.1 -0.1
0.9
1.9
(𝑥 𝑖 − 𝜇)2
1.21 0.01 0.81
3.61
(𝑥 𝑖 − 𝜇)2 𝑝 𝑖
0.363 0.004 0.162 0.361
𝜎 2 = (𝑥 𝑖 − 𝜇 𝑥 )2 ∗ 𝑝 𝑖 = 0.89
Standard Deviation = 𝜎 = 0.89 = 0.943
= 𝟏. 𝟏

4. Rules for Means and Variances of Random Variables
When you add or subtract 2 random variables, what happens to their mean and variance?
Rules: 𝜇 𝑥+𝑦 = 𝜇 𝑥 + 𝜇 𝑦
𝜇 𝑥−𝑦 = 𝜇 𝑥 − 𝜇 𝑦
𝜎2 = 𝜎2 + 𝜎2
𝑥+𝑦
𝑥
𝑦
𝜎2 = 𝜎2 + 𝜎2
𝑥−𝑦
𝑥
𝑦
When you multiply each outcome of a random variable by a number, what happens to the
mean and variance?
Rules: 𝜇 𝑎𝑥 = 𝑎𝜇 𝑥
𝜎 2 = 𝑎2 𝜎 2
𝑎𝑥
𝑥
Note: Variance is multiplied by a squared factor since the spread is being multiplied in 2 directions.
When you add a number to each outcome of a random variable, what happens to the mean
and the variance?
Rules: 𝜇 𝑏+𝑋 = 𝑏 + 𝜇 𝑥
𝜎2 = 𝜎2
𝑥
𝑏+𝑋
Note: Variance doesn’t change because the spread of the data remains the same