Question

y x is normal variate such that mx(t)=e³+8+2,then the varience is 

• A. 16
• B. 8
• C. 64
• D. 4

Answer 1

The Correct Option is A

We need to determine the variance of a normal variate \( X \) given its moment-generating function (MGF) \( M_X(t) = e^{3t + 8t^2} \).

1. Understanding the Moment-Generating Function:
For a normal random variable \( X \sim N(\mu, \sigma^2) \), the MGF is:

\( M_X(t) = e^{\mu t + \frac{1}{2} \sigma^2 t^2} \)
where \( \mu \) is the mean and \( \sigma^2 \) is the variance. We are given \( M_X(t) = e^{3t + 8t^2} \), which we can compare to the standard form.

2. Comparing the Given MGF:
The given MGF is:

\( M_X(t) = e^{3t + 8t^2} \)
Matching this with the standard form \( e^{\mu t + \frac{1}{2} \sigma^2 t^2} \), we identify the exponents:
- The term with \( t \): \( \mu t = 3t \), so \( \mu = 3 \).
- The term with \( t^2 \): \( \frac{1}{2} \sigma^2 t^2 = 8t^2 \), so \( \frac{1}{2} \sigma^2 = 8 \).

3. Solving for Variance:
From \( \frac{1}{2} \sigma^2 = 8 \), multiply both sides by 2:

\( \sigma^2 = 16 \)
The variance of \( X \) is \( \sigma^2 \).

Final Answer:
The variance of the normal variate \( X \) is \( 16 \).