Question

Let the moment generating function (m.g.f) of a random variable is of the form \(M_X(t)=(0.4e^t +0.6)^8\). Find the mean and variance of a random variable Y=3X+2.

• A. E(Y) =9.4 and V(Y) = 17.28
• B. E(Y) =9.6 and V(Y) = 17.26
• C. E(Y) =9.4 and V(Y) = 17.26
• D. E(Y) =9.6 and V(Y) = 17.28

Hint:

When applying linear transformations to variance, always square the multiplier:
\[ V(aX + b) = a^2 V(X) \]
Since \(a = 3\), the variance must be multiplied by \(9\).
\(9 \times 1.92 = 17.28\), which immediately eliminates options (B) and (C).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The moment generating function (MGF) uniquely determines the probability distribution of a random variable.
The given MGF has the structure of a Binomial distribution:
\[ M_X(t) = (q + p \cdot e^t)^n \]
Using the properties of expectation and variance, we can find the mean and variance of any linear transformation \(Y = aX + b\).
Key Formula or Approach:
For a Binomial distribution \(X \sim \text{Binomial}(n, p)\):
\[ E(X) = n \cdot p \]
\[ V(X) = n \cdot p \cdot q \]
For a linear transformation \(Y = aX + b\):
\[ E(Y) = a \cdot E(X) + b \]
\[ V(Y) = a^2 \cdot V(X) \]

Step 2: Detailed Explanation:
First, identify the parameters of the distribution of \(X\) from its MGF \(M_X(t) = (0.4e^t + 0.6)^8\):
- Number of trials, \(n = 8\) (note: the exponent in the printed question was rendered with a typo as "l", but mathematically corresponds to 8)
- Probability of success, \(p = 0.4\)
- Probability of failure, \(q = 0.6\)
Calculate the mean and variance of \(X\):
\[ E(X) = n \cdot p = 8 \cdot 0.4 = 3.2 \]
\[ V(X) = n \cdot p \cdot q = 8 \cdot 0.4 \cdot 0.6 = 1.92 \]
Now, apply the linear transformation rules for the variable \(Y = 3X + 2\):
Calculate the variance of \(Y\):
\[ V(Y) = 3^2 \cdot V(X) = 9 \cdot 1.92 = 17.28 \]
Calculate the expected value of \(Y\):
If we evaluate the expectation strictly as written:
\[ E(Y) = 3 \cdot E(X) + 2 = 3(3.2) + 2 = 9.6 + 2 = 11.6 \]
However, the official question options write the mean as \(E(Y) = 9.6\), which corresponds to the unshifted linear combination \(Y = 3X\) (omitting the additive constant \(+2\) in the mean calculation by typographical oversight).
Comparing the computed variance \(V(Y) = 17.28\) and the closest matching option, we find that Option (D) represents the intended key.

Step 3: Final Answer:
The correct option is (D).