Final answer:
To find the moment generating function (MGF) for a discrete random variable, use the formula M(t) = E(e^(tx)). Calculate M(t) using the given probabilities for X, and then differentiate it to find E[Xj] for j = 1, 2, 3, 4.
Step-by-step explanation:
To find the moment generating function (MGF) for a discrete random variable, we need to use the formula M(t) = E(e^(tx)), where E is the expected value and x is the random variable. For each value of x, we multiply it by e^(tx) and the corresponding probability, then sum up all the products.
In this case, the MGF for the random variable X can be calculated as follows:
M(t) = 0.10 * e^(-2t) + 0.25 * e^(-t) + 0.30 * e^(0) + 0.25 * e^t + 0.10 * e^(2t)
Now, to find E[X^j] for j = 1, 2, 3, 4, we differentiate the MGF with respect to t j times and then evaluate it at t = 0. Let's calculate each expectation:
E[X] = M'(0) = -2 * 0.10 * e^(-2*0) - 1 * 0.25 * e^(-1*0) + 0 * 0.30 * e^(0*0) + 1 * 0.25 * e^(1*0) + 2 * 0.10 * e^(2*0) = 0
E[X^2] = M''(0) = (-2)^2 * 0.10 * e^(-2*0) + (-1)^2 * 0.25 * e^(-1*0) + 0^2 * 0.30 * e^(0*0) + 1^2 * 0.25 * e^(1*0) + 2^2 * 0.10 * e^(2*0) = 0.10 + 0.25 + 0.10 = 0.45
E[X^3] = M'''(0) = (-2)^3 * 0.10 * e^(-2*0) + (-1)^3 * 0.25 * e^(-1*0) + 0^3 * 0.30 * e^(0*0) + 1^3 * 0.25 * e^(1*0) + 2^3 * 0.10 * e^(2*0) = -0.20 - 0.25 + 0.80 = 0.35
E[X^4] = M''''(0) = (-2)^4 * 0.10 * e^(-2*0) + (-1)^4 * 0.25 * e^(-1*0) + 0^4 * 0.30 * e^(0*0) + 1^4 * 0.25 * e^(1*0) + 2^4 * 0.10 * e^(2*0) = 0.40 + 0.25 + 0.40 = 1.05