Question

Recall that the moment generating function of a random variable X is M X(t)= E(exp(tX))

(a) Prove that for a Gaussian RV X∼N(μ,σ 2), its mgf is M X​ (t)=exp(μt+ 2σ 2t 2). (Hint: Use the fact that ∫ −[infinity][infinity]​exp(− 2b 2(x−a) 2)dx=b 2π for a,b∈R,b>0, for a suitable pair a,b.

Answer 1

To prove the given moment generating function for a Gaussian random variable with mean μ and variance σ2, we use the integral formula provided with suitable selections for a and b, substituting the probability density function into the expected value of μtX, simplifying, and computing the integral.

Step-by-step explanation:

The question asks to prove that the moment generating function (mgf) of a Gaussian random variable X with parameters μ (mean) and σ2 (variance) is given by MX(t) = exp(μt + σ2t2). Leveraging the hint, we can select appropriate values of a and b to simplify the integral for the mgf. If X follows a normal distribution X ~ N(μ, σ2), then its probability density function is f(x) = (1/(σ√(2π)))·exp(-((x - μ)2) / (2σ2)). The mgf is the expected value of exp(tX), which we can compute using its density function over the entire range of X as:

MX(t) = ∫ exp(tx)·f(x)dx.

Substituting the density function and simplifying the integral, we will find that a equals μ and b equals σ, which allows us to use the given integral formula to compute the mgf, thereby confirming the answer. This process involves completing the square in the exponent and performing a change of variables to relate it to the formula provided in the hint.