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Distribution of Functions – Moment Generating Function (MGF)
Step 2: MGF of the Sum S = X 1 + X 2 + X 3
Since X 1 , X 2 , X 3 are independent, the MGF of their sum is the product of their
MGFs:
MS (t) = MX 1 (t) · MX 2 (t) · MX 3 (t) =
2 − t
Step 3: Identify the Distribution from the MGF
Recall that the MGF of a Gamma distribution with shape r and rate λ is:
MY (t) =
λ
λ − t
r
, for t < λ
Comparing:
MS (t) =
2 − t
We recognize this as the MGF of a Gamma distribution with shape r = 3, rate
λ = 2.
Final Answer
X 1 + X 2 + X 3 ∼ Gamma(r = 3, λ = 2)
Sam Maseno University August 19, 2025 105 / 132
Distribution of Functions – Moment Generating Function (MGF)
Using the MGF to Identify the Distribution of a Linear Transformation of a Normal Variable Problem Let X ∼ N (1, 4). Define a new random variable Y = 2X − 3. Find the moment generating function (MGF) of Y and identify its distribution. Solution Step 1: MGF of a Normal Distribution The MGF of a normal random variable X ∼ N (μ, σ 2 ) is:
MX (t) = exp
μt +
σ 2 t 2
In our case, μ = 1, σ 2 = 4, so:
MX (t) = exp
t + 2t 2
Step 2: MGF of the Transformed Variable Y = 2X − 3 We use the rule for linear transformations of MGFs: If Y = aX + b, then: MY (t) = e bt · MX (at)
Here, a = 2, b = − 3 , so:
MY (t) = e − 3 t · MX (2t) = e − 3 t · exp
(2t) + 2(2t) 2
= e − 3 t · exp
2 t + 8t 2
= exp
− 3 t + 2t + 8t^2
= exp
−t + 8t^2
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