Question

Let \( X \) be a discrete random variable with the moment generating function \[ M_X(t) = \frac{(1 + 3e^t)^2 (3 + e^t)^3}{1024}, t \in \mathbb{R}. \] Then

• A. \( E(X) = \frac{9}{4} \)
• B. \( \text{Var}(X) = \frac{15}{32} \)
• C. \( P(X \geq 1) = \frac{27}{1024} \)
• D. \( P(X = 5) = \frac{3}{1024} \)

Hint:

To find the variance of a discrete random variable, differentiate the moment generating function twice and evaluate at \( t = 0 \).

Answer 1

The Correct Option is A

Step 1: Expand the MGF

$$M_X(t) = \frac{1}{1024}(1 + 3e^t)^2(3 + e^t)^3$$

Recognize that $1024 = 4^5 = 2^{10}$.

Rewrite: $$M_X(t) = \frac{1}{4^2 \cdot 3^3}(1 + 3e^t)^2(3 + e^t)^3$$

$$= \left(\frac{1 + 3e^t}{4}\right)^2 \left(\frac{3 + e^t}{4}\right)^3$$

$$= \left(\frac{1}{4} + \frac{3e^t}{4}\right)^2 \left(\frac{3}{4} + \frac{e^t}{4}\right)^3$$

This is the MGF of the sum of 5 independent Bernoulli random variables:

• Two with $P(Y_i = 1) = 3/4$
• Three with $P(Z_j = 1) = 1/4$

So $X = Y_1 + Y_2 + Z_1 + Z_2 + Z_3$ where all are independent.

Step 2: Calculate $E[X]$

$$E[X] = 2 \cdot \frac{3}{4} + 3 \cdot \frac{1}{4} = \frac{6}{4} + \frac{3}{4} = \frac{9}{4}$$

Option (A) is TRUE 

Step 3: Calculate $Var(X)$

$$Var(X) = 2 \cdot \frac{3}{4} \cdot \frac{1}{4} + 3 \cdot \frac{1}{4} \cdot \frac{3}{4}$$

$$= 2 \cdot \frac{3}{16} + 3 \cdot \frac{3}{16} = \frac{6}{16} + \frac{9}{16} = \frac{15}{16}$$

Option (B) is FALSE (it claims $Var(X) = 15/32$) 

Step 4: Calculate $P(X \geq 1)$

$$P(X \geq 1) = 1 - P(X = 0) = 1 - P(\text{all 5 variables are 0})$$

$$= 1 - \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{3}{4} \cdot \frac{3}{4} \cdot \frac{3}{4}$$

$$= 1 - \frac{27}{1024} = \frac{997}{1024}$$

Option (C) is FALSE (it claims $P(X \geq 1) = 27/1024$) 

Step 5: Calculate $P(X = 5)$

$$P(X = 5) = P(\text{all 5 variables are 1})$$

$$= \frac{3}{4} \cdot \frac{3}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} = \frac{9 \cdot 1}{1024} = \frac{9}{1024}$$

Option (D) is FALSE (it claims $P(X = 5) = 3/1024$) 

Answer: (A) $E(X) = \frac{9}{4}$