Question

Q2. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Let b> 0. (a) Find the cumulative distribution function of Y = XI(X < b} (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.

Answer 1

$(a)~CDF~of~Y~is\\ F_Y(y)=P(Y\leq y)=P(X\leq y|X\leq b)=\frac{P(X\leq y)}{P(X\leq b)}=\left\{\begin{matrix} 0~if~y<0\\ \frac{F_X(y)}{F_X(b)}~if~y<b\\ 1~if~y\geq b \end{matrix}\right.\\ (b)~X\sim exponential~with~parameter~\theta\\ Then~F_X(x)=\int_0^x\lambda e^{-\lambda x}dx=1-e^{-\lambda x},~x>0\\ CDF~of~Y~is\\ F_Y(y)=\left\{\begin{matrix} 0~if~y<0\\ \frac{1-e^{-\lambda y}}{1-e^{-\lambda b}}~if~y<b\\ 1~if~y\geq b \end{matrix}\right.$