# distribution of random variable X

Random variable $X$ has exponential distribution $\varepsilon(\lambda)$ with the probability of $0.3$, and with the probability of $0.7$ it has a distribution whose density function is equal to $f_2(x)=1/2 e^{-|x+1|}$, for every $x \in R$. Find the density function and mathematical expectation of the random variable $X$.

I know what the exponential distribution looks like, but I just don't understand how to use any of the given information in order to solve the problem.. Any help would be really appreciated :)

• You can get the pdf by just adding the two pdfs multiplied by the corresponding weights. The expectation is then just an integral problem.
– Ian
May 12 '17 at 13:27

Let $f_1(.)$ and $\mu_1$ be the density function and mean of the exponential distribution. Also let $\mu_2$ be the mean of the distribution with density $f_2(.)$. Then density of $X$ is $$0.3f_1(x)+0.7f_2(x)$$ and mean of $X$ is $$0.3\mu_1+0.7\mu_2$$