Question

If Y is a random variable with moment-generating function m(t) and if W is given by W=aY+b, show that the moment-generating function of W is eᵗᵇ m(at).

Answer 1

The moment-generating function of W, when W = aY + b, is shown to be e^tb m(at) by using the definition of MGF, substituting the expression for W, and then factoring out the constant e^tb while recognizing that e^atY is the MGF of Y evaluated at at.

Step-by-step explanation:

To demonstrate that the moment-generating function of W is etb m(at), when W is defined as W = aY + b, we first need to recall the definition of the moment-generating function (MGF). The MGF of a random variable Y, denoted m(t), is given by E(etY), where E indicates the expected value. Now, for the random variable W:

1. Compute the MGF of W by taking the expected value of etW.
2. Replace W with its expression in terms of Y: et(aY + b).
3. Separate the exponential into a product: eatYetb.
4. The etb is a constant with respect to Y and can be factored out of the expectation.
5. The remaining component eatY is the MGF of Y, evaluated at at instead of t.

Hence, the MGF of W is etb m(at), with m(at) being the MGF of Y evaluated at at.