Let 𝑋∼𝖤𝗑𝗉(𝜆), 𝜆>0. What is the moment generating function (MGF) 𝑀𝑋(𝑡) of 𝑋?

Question

exp(−𝜆𝑡)

𝜆𝜆−𝑡∀𝑡∈ℝ

1𝜆

−𝜆𝑡−𝜆∀𝑡<𝜆

Answer 1

The moment generating function of 𝑋∼𝖤𝗑𝗉(𝜆) is 𝑀𝑋(𝑡) = 1/(1-𝜆𝑡), 𝑡<1/𝜆.

To see this, we can use the definition of the moment generating function:

𝑀𝑋(𝑡) = 𝔼[𝑒^𝑡𝑋]

= ∫₀^∞ e^𝑡𝑥 𝜆e^−𝜆𝑥 dx (since 𝑋 has an exponential distribution with parameter 𝜆)

= 𝜆 ∫₀^∞ e^(𝑡-𝜆)𝑥 dx

= 𝜆/(𝜆-𝑡), 𝑡<𝜆.

Therefore, the moment generating function of 𝑋 is 1/(1-𝜆𝑡), 𝑡<1/𝜆.