is an alternative specification of a real-valued random variable’s probability distribution
Not all random variables have an MGF
As its name implies, the MGF can be used to compute a distribution’s moments: the 𝑛^th moment about 0 is the 𝑛^th derivative of the MGF evaluated at 0

MGF - Definition

The MGF of a random variable 𝑋 is defined as:

$M_{X} (t) = E [e^{tX}] = {\sum_{x \in X} e^{t x} P_{X} (x) \int_{- \infty}^{\infty} e^{t x} f_{X} (x) d x X is discrete X is continuous$

Provided the expectation 𝐄 exists for some 𝑡 in a neighborhood of 0.

MGF - Using MGF to Calculate Moments

The 𝑛th moment of random variable 𝑋 denoted as 𝐄[𝑋^𝑛] is defined as the 𝑛^th derivative of the MGF of 𝑋 evaluated at 𝑡=0:

If the MGF of 𝑋 is 𝑀_𝑋(𝑡), then the MGF of 𝑎𝑋+𝑏 is 𝑒^𝑏𝑡𝑀_𝑋(𝑎𝑡)