# Continuous Random Variables including exponential distribution

Let $X$ be an exponential random variable with parameter $\lambda=9$. Let $Y$ be the random variable defined by $Y=10e^X$. Compute the probability density function of $Y$: what is $f_Y(t)$ (for $t\geq10)$?

• Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. – José Carlos Santos Mar 15 '18 at 20:01
• I assume you mean $Y=10e^X$. Please make an effort to compute $P[Y\leq y]$. – Michael Mar 15 '18 at 20:26

Here are some hints to get you started.

The cdf of $Y$ is given by

\begin{align*} F_Y(t) &=P(Y\leq t)\\\\ &=P(10e^X \leq t) \end{align*}

Can you go from here to get this in terms of the cdf of $X$? That is, $P(X\leq x)$.

• I calculated for a few times but I am not sure whether this is correct: 10e^x <= t ln10+x <= lnt x <= lnt-ln10 – Qihong Dai Mar 16 '18 at 0:31