Question
A random variable $X$ has the exponential distribution$$f(x)=\lambda e^{-\lambda x}, \quad x>0$$Show that the moment-generating function of $X$ is$$M_{X}(t)=\left(1-\frac{t}{\lambda}\right)$$(b) Find the mean and variance of $X$.
Step 1
For the exponential distribution, we can write this as an integral: $$ M_X(t) = \int_0^\infty e^{tx} \lambda e^{-\lambda x} dx $$ Show more…
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