How to write down the density, the expected value and the variance of a random variable distributed as an exp(γ)?

Question

So for the density is it just the probability density function or there is something more? I'm not exactly sure what I'm asked for. Same for exp and variance

Answer 1

Do you mean an exponential random variable $X$ with parameter $\gamma$?

I think the first question is just to write the probability density function, that is, as you can see on Wikipedia, $$ f(x) = \begin{cases} \gamma e^{-\gamma x}, & \text{if } x \ge 0, \\ 0, & \text{if } x < 0. \end{cases} $$ To compute the expected value and the variance exploit the usual formulas $$ \mathbb{E}[X] = \int_\mathbb{R} x\, f(x) \, dx $$ $$ \mathbb{E}[X^2] = \int_\mathbb{R} x^2\, f(x) \, dx $$ and $$ \textrm{Var} (X) = \mathbb{E}[X^2]-\mathbb{E}[X]^2 $$ to get $\mathbb{E}[X] = \frac{1}{\gamma}$ and $\textrm{Var}(X) = \frac{1}{\gamma^2}$.