Question

Let X1, X2 be a random sample drawn from some normal distribution with mean 1 and variance 2. Find the moment generation function of Y = (X1 − X2)/2.

Answer 1

The moment generation function of Y = (X1 − X2)/2 is M_Y(t) = e^t * e^(t^2/2).

Step-by-step explanation:

The moment generation function of a random variable Y can be found by taking the expected value of e^(tY), where t is a parameter. In this case, we want to find the moment generation function of Y = (X1 − X2)/2.

Step 1: Start with the definition of the moment generation function: M_Y(t) = E(e^(tY)).

Step 2: Substitute the expression for Y: M_Y(t) = E(e^(t((X1 − X2)/2))).

Step 3: Use the linearity of expectation: M_Y(t) = E(e^((t/2)X1) * e^(-(t/2)X2)).

Step 4: Since X1 and X2 are independent, we can split the expectation: M_Y(t) = E(e^((t/2)X1)) * E(e^(-(t/2)X2)).

Step 5: The moment generating function of a normal random variable X with mean μ and variance σ^2 is e^(μt + (σ^2t^2)/2). Thus, the moment generating function of X1 is e^((t/2)*1 + (2t^2)/8) = e^(t/2 + t^2/4) and the moment generating function of X2 is e^((t/2)*1 + (2t^2)/8) = e^(t/2 + t^2/4).

Step 6: Substitute these values back into the expression: M_Y(t) = e^(t/2 + t^2/4) * e^(t/2 + t^2/4) = e^t * e^(t^2/2).

Therefore, the moment generation function of Y = (X1-X2)/2 is M_Y(t) = e^t * e^(t^2/2).