Identifying the MGFs | Binomial vs Poisson | GATE ST 2025 | Problem-58 | RitwikMath

Crack **moment generating functions**! Given MGFs of independent X, Y, Z, find **10 × P(X Y + Z)**.

**Distributions identified:**
- \(M_X(t) = \frac{2 + e^t}{3^2} \implies X \sim \text{Binomial}(2, \frac{1}{3})\)
\(P(X=0) = \frac{4}{9}, P(X=1) = \frac{4}{9}, P(X=2) = \frac{1}{9}\)
- \(M_Y(t) = e^{e^t - 1} \implies Y \sim \text{Poisson}(1)\)
- \(M_Z(t) = e^{2(e^t - 1)} \implies Z \sim \text{Poisson}(2)\)
- \(S = Y + Z \sim \text{Poisson}(3)\)

**Key calculation:**
\[
P(X S) = \sum_{k=0}^2 P(X=k)P(S k) = \frac{4}{9}e^{-3} + \frac{1}{9} \cdot 4e^{-3} = \frac{8}{9}e^{-3}
\]
\[
10 \times P(X S) = \frac{80}{9}e^{-3} \approx 0.44
\]

Perfect for mastering **MGF recognition** and **discrete probability calculations**!

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