Random Variables Probability Continued Chapter 7 Part 2

Developing Transformation Rules Consider the following distribution for the random variable X:

X+1 What is the probability distribution for X+1?

2 X What is the probability distribution for 2 X?

Consider Suppose that E(X) = 2. 5, Var(X) = 0. 2 What is E(X+5) = ? , Var(X+5) = ? E(X+5) = 7. 5, Var(X+5) = 0. 2 What is E(X – 2. 2) = ? , Var(X – 2. 2) = ? E(X – 2. 2) = 0. 3, Var(X – 2. 2) = 0. 2 What is E(3 X) = ? , Var(3 X) = ? E(3 X) = 7. 5, Var(3 X) = 1. 8 What is E(2 X – 1) = ? , Var(2 X – 1) = ? E(2 X – 1) = 4, Var(2 X – 1) = 0. 8

Rule 1: If X is a random variable and a and b are fixed numbers, then ma + b. X = a + bm. X Rule 1: If X is a random variable and a and b are fixed numbers, then s 2 a + b. X =b 2 s 2 X

X+X What is the probability distribution for X+X? . 2 . 8 1 . 2 1 . 8 2 . 2 1 2. 8 2

X+X

X–X What is the probability distribution for X–X ? . 2 . 8 1 . 2 1 . 8 2 . 2 1 2. 8 2

X–X

Consider Suppose that E(X) = 2. 5, Var(X) =. 16, E(Y) = 1. 2, Var(Y) =. 36 What is E(X+Y) = ? , Var(X+Y) = ? E(X+Y) = 3. 7, Var(X+Y) =. 52 E(X – Y) = ? , Var(X – Y) = ? E(X – Y) = 1. 3, Var(X – Y) =. 52 What is s(X) = ? , s(Y) = ? , s(X+Y) = ? s(X) =. 4, s(Y) =. 6, s(X+Y) =. 7211

Rule 2: If X and Y are random variables, then m. X + Y = m. X + m. Y m. X – Y = m. X – m. Y Rule 2: If X and Y are independent random variables, then s 2 X + Y = s 2 X + s 2 Y s 2 X - Y = s 2 X + s 2 Y Note: cannot combine standard deviation directly.