Question

Show that if X ~ N(μ, σ²), then N_X(t) = exp(μt + t²σ²/2) for all t in R.

Answer 1

To show that if X ~ N(μ, σ²), then N_X(t) = exp(μt + t²σ²/2) for all t in R, we can use moment generating functions.

Step-by-step explanation:

To show that if X ~ N(μ, σ²), then N_X(t) = exp(μt + t²σ²/2) for all t in R, we can make use of moment generating functions.

1. Start by finding the moment generating function (MGF) of X, which is given as M(t) = E[e^(tX)].
2. Next, using the properties of MGF, we can calculate M_X(t) = M(t - μ) for the standardized random variable X.
3. Finally, substituting the values of X and the MGF into the equation, we can see that N_X(t) = exp(μt + t²σ²/2) for all t in R.