Question

Determine the value c so that each of the following functions can serve as a probability distribution of the discrete random variable X:

a) for = 0,1,2,3,4,5,6,7
b) for = 0,1,2,3,4

Answer 1

To determine the value of c for which the functions can serve as a probability distribution, sum all the probabilities multiplied by c and set this sum equal to 1. Then solve for c to ensure the sum of all probabilities equals 1.

Step-by-step explanation:

The student has asked to determine the value of c for which the given functions can serve as a probability distribution of a discrete random variable X. To satisfy the conditions of a probability distribution for a discrete random variable, two criteria must be met:

• Each probability P(X=x) must be between 0 and 1 inclusive.
• The sum of all probabilities must equal 1.

To find the value of c, we would set up the equation:

\( c \times P(X=0) + c \times P(X=1) + ... + c \times P(X=n) = 1 \)

Where \( P(X=x) \) are the probabilities for each value x that the random variable X can take, and n is the maximum value X can take (such as 7 or 4 in the given examples).

Once all the probabilities are given, we solve this equation for c to ensure the probabilities add up to 1.