Question

Show that if X ~ N(μ, σ²), then E(X) = (∂M_X(t)/∂t)|_(t=0). Hint: E(X) = μ.

Answer 1

To show that if X ~ N(μ, σ²), then E(X) = (∂M_X(t)/∂t)|_(t=0), we can use the moment generating function (MGF) for a normal distribution.

Step-by-step explanation:

To show that if X ~ N(μ, σ²), then E(X) = (∂M_X(t)/∂t)|_(t=0), we can start by using the moment generating function (MGF) for a normal distribution.

1. The MGF of X is M_X(t) = e^(μt + σ²t²/2).
2. Now, let's find the derivative of M_X(t) with respect to t: ∂M_X(t)/∂t = (μ + σ²t)e^(μt + σ²t²/2).
3. Finally, plug in t=0 and we get: (∂M_X(t)/∂t)|_(t=0) = μ. Therefore, E(X) = μ.