Question

If a discrete random variable X follows the uniform distribution and assumes only the values 8, 9, 11, 15, 18, and 20, then P(|X -14| < 5) is

• A. \(\frac{1}{2}\)
• B. \(\frac{1}{5}\)
• C. \(\frac{1}{4}\)
• D. \(\frac{2}{3}\)

Answer 1

The Correct Option is A

If the discrete random variable \(X\) follows a uniform distribution and assumes the values 8, 9, 11, 15, 18, and 20, then we need to calculate \(P(|X - 14| < 5)\). This problem can be solved as follows:

First, express the inequality \(|X - 14| < 5\): 

\(-5 < X - 14 < 5\)

By adding 14 to all parts of the inequality, we obtain:

\(9 < X < 19\)

Thus, we are looking for values of \(X\) within the range from 10 to 18 inclusive (since \(X\) can only assume integer values).

The values of \(X\) that satisfy this are 11, 15, and 18.

Next, determine how many values satisfy this condition. There are 3 values: 11, 15, and 18.

Calculate the probability:

The uniform distribution over 6 values (8, 9, 11, 15, 18, and 20) means each has a probability of \(\frac{1}{6}\).

Thus, the probability \(P(|X - 14| < 5)\) is given by:

\[P(X = 11) + P(X = 15) + P(X = 18) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\]

Therefore, \(P(|X - 14| < 5)\) is \(\frac{1}{2}\).