Problem 7. The mgf and the mean associated with a discrete random variable X are given by Mx(t) = ael +be' "D, E[X] = 3. Find: (a) the scalar parameters a and b; (b) P(X =1) and E[X2]. () P(X +Y = 2), where Y is a random variable that is independent of X and is identically distributed to X. To find the scalar parameters a and b for the given moment generating function (mgf) Mx (t) = aet + belle-!) and the mean E[ X] - 3, we need to follow these steps: 1. Calculate the mean from the mgf: The mean E[ X ] is the first derivative of the mof evaluated at t = 0: E[X] - M'x(0) 2. Find the first derivative of the mgf: Mx(t) = aet + bet(e'-1) Mx(t) = a- At 4(e'-1) dt dt M'x(t) = aet + b. et(e'-1) . 4et M'x(t) = aet + 4bel('-1) et3. Evaluate the derivative at t = 0: M'x(0) = ae" + 4be4 ("-1)20 M'(0) = a + 4be M'(0) = a + 46 4. Set the expression equal to the given mean: a + 4b - 3 To find a and b, we need another equation. Given that the mof at t = 0 should equal 1 (since Mx(0) - E[e0* ] = 1): Mx(0) = ae + belle"-1) - atb=1 Now we have a system of linear equations: 1. a + 4b - 3 2. atb =1To find a and b, we need another equation. Given that the mof at t = 0 should equal 1 (since Mx(0) - Ele"* ] = 1): Mx(0) = ae" + bet("-1) - atb=1 Now we have a system of linear equations: 1. a + 4b = 3 2. atb =1 Solving this system: Subtract the second equation from the first: (a + 4b) - (atb) -3-1 3b = 2 b =Substitute b back into the second equation: a+ = 1 a = 1 a = Therefore, the scalar parameters are: a b =