Question

A discrete random variable X has the following probability distribution.

X 0 1 2 3
P(X = x) 0.475 2k k 0.1 - 6k
a. Find the value of k. (open-ended) b. Find the expected value of X

Answer 1

The value of k in the given probability distribution is 0.141666..., and the expected value of X is 0.566666....

Step-by-step explanation:

To determine the value of k, we need to use the property that the sum of all probabilities in a discrete probability distribution must equal 1. The given probability distribution is:

• P(X=0) = 0.475
• P(X=1) = 2k
• P(X=2) = k
• P(X=3) = 0.1 - 6k

Summing these probabilities and setting them equal to 1 gives us the equation:

0.475 + 2k + k + (0.1 - 6k) = 1

Solving for k, we combine like terms and get:

3k - 6k + 0.575 = 1

-3k + 0.575 = 1

-3k = 0.425

k = -0.425 / -3 = 0.141666...

To find the expected value (E(X)) of this distribution, we multiply each value of the random variable by its respective probability and sum these products:

E(X) = 0(0.475) + 1(2k) + 2(k) + 3(0.1 - 6k)

Substituting in our found value of k:

E(X) = 0 + 2(0.141666...) + 2(0.141666...) + 3(0.1 - 6(0.141666...))

Then, simplifying:

E(X) = 2(0.141666...) + 2(0.141666...) + 3(0.1 - 0.85)

E(X) = 0.283333... + 0.283333... + 0

E(X) = 0.566666...