Consider the following the continuous RVs X1, X2, Y, and Z: X1 ~ U(10, 20), X2 ~ U(10, 20), Y = X1 + X2, Z = Y + 20, a. What is the MGF of Z? b. Draw the pdf of Z. c. Use the MGF of Z to get its mean. d. Find P( Z < 50). Shade the appropriate area that corresponds to the said probability
I have arrived at an answer for A but I can't figure out the distribution which makes it hard for me to answer the following questions.
If it were given that $X_1,X_2$ are independent, then you can use the basic result on the moment-generating function of the sum of two independent random variables.
Next you have \begin{align} & M_{Y+20} (t) =\operatorname{E}(e^{t(Y+20)}) = \operatorname{E}(e^{tY} e^{20t}) \\[10pt] = {} & e^{20t} \operatorname{E}(e^{tY}) \text{ because $e^{20t}$ is constant, i.e. not random} \\[10pt] = {} & e^{20t} M_Y(t). \end{align}