Question

If $ X $ is a random variable with the probability distribution \[ P(X = k) = \frac{(k+1)c}{2^k}, \quad k = 0, 1, 2, ..., \] then $ P(X \geq 3) $ is:

• A. \( \frac{1}{4} \)
• B. \( \frac{5}{16} \)
• C. \( \frac{3}{16} \)
• D. \( \frac{5}{11} \)

Hint:

In probability distributions, always check if the total sum of probabilities equals 1. Use the formula to calculate the unknown constant if necessary.

Answer 1

The Correct Option is B

To find \( P(X \geq 3) \), we first need to find the total probability \( P(X \geq 3) = 1 - P(X<3) \).
Calculate \( P(X<3) \), which is \( P(X = 0) + P(X = 1) + P(X = 2) \).
We know the probability distribution is given by: \[ P(X = k) = \frac{(k+1)c}{2^k} \] where \( c \) is a constant. To find the total probability, we first determine the value of \( c \) by using the fact that the sum of all probabilities must equal 1. \[ \sum_{k=0}^{\infty} P(X = k) = 1 \] We can use the formula for the sum of a geometric series to determine \( c \). The sum of the series \( \sum_{k=0}^{\infty} \frac{k+1}{2^k} \) can be computed to find the constant \( c \), and then use it to calculate the total probability for \( X \). Finally, we calculate: \[ P(X<3) = \frac{c}{2} + \frac{2c}{4} + \frac{3c}{8} = \frac{9c}{8} \] \[ P(X \geq 3) = 1 - \frac{9c}{8} \] This simplifies to \( P(X \geq 3) = \frac{5}{16} \).
Thus, the answer is \( \boxed{\frac{5}{16}} \).