Question

Let X and Y be independent random variables, with known moment generang functions Mx(t) and My (t) and Z be such that P(Z = 1) = 1-P(Z 0) = p E (0,1). Compute the moment generating function of the random variable S- ZX (1 - Z)Y. [The distribution of S is called a mirture of the distributions of X and Y.] Your answer can be left in terms of Mx(t) and My (t) Hint: If you don't know how/where to start, think about conditioning in this case, condi tioning on Z looks promising)

Answer 1

$MGF~of~S=M_S(t)=E(e^{St})=E(e^{ZXt+(1-Z)Yt})\\ =E\left(E(e^{ZXt+(1-Z)Yt}|Z) \right )=E(E(e^{ZtX}|Z)E(e^{(1-Z)tY}|Z))\\ since~X~and~Y~are~independent~so~X|Z~and~Y|Z~are~also~independent.\\ =E(M_X(tZ)M_Y((1-Z)t))=M_X(t)M_Y(0*t)p+M_X(t*0)M_Y(t)(1-p)\\ =M_X(t)\times p+M_Y(t)\times (1-p),~since~M_X(0)=M_Y(0)=1$