This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — The given moment generating function (MGF) is m(t) = e^3(e^t-1). This is the MGF of a Poisson distribution with parameter . The standard MGF for a Poisson distribution is M_X(t) = e^(e^t-1). By comparing the given MGF with the standard form, we can identify that = 3. Thus, X follows a Poisson distribution with parameter = 3. For a Poisson distribution X Pois(): • The probability mass function is P(X=k) = ^k e^-k! • The first moment (mean) is E[X] = • The second moment is E[X^2] = + ^2 • The variance is Var(X) = a) Find P(X=0) Step 1: Use the Poisson probability mass function with = 3 and k=0. P(X=0) = 3^0 e^-30! Step 2: Simplify the expression. P(X=0) = 1 · e^-31 = e^-3 ≈ 0.049787 The probability P(X=0) is 0.0498. b) Find the first and second moments of X Step 1: Calculate the first moment, E[X]. For a Poisson distribution, E[X] = . E[X] = 3 Step 2: Calculate the second moment, E[X^2]. For a Poisson distribution, E[X^2] = + ^2. E[X^2] = 3 + 3^2 = 3 + 9 = 12 The first moment is 3 and the second moment is 12. c) Find Var(X) Step 1: Use the formula for variance, Var(X) = E[X^2] - (E[X])^2. Using the results from part (b): Var(X) = 12 - (3)^2 = 12 - 9 = 3 Alternatively, for a Poisson distribution, Var(X) = . Var(X) = 3 The variance Var(X) is 3. What's next?