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Suppose X has an exponential distribution with $E[X] = 3$. Suppose Y has a uniform distribution on $[0, 5]$. Suppose X and Y are independent.

  • (a) Find $P(X≤1)$.
  • (b) Find the joint density of X and Y .

I know exponential pdf is $\lambda e^{-\lambda x}$, but I don't know how to get $\lambda$ in computing $\int_{0}^{1} \lambda e^{-\lambda x} dx$

I don't know how to start (b). Please help.

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closed as off-topic by Chappers, GNUSupporter 8964民主女神 地下教會, Namaste, JMP, Claude Leibovici Jan 12 '18 at 8:11

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Hints.

(a) The exponential distribution with rate $\lambda$ has an expected value of $1/\lambda$.

(b) Since $X$ and $Y$ are independent, if they have PDFs $f_X(x)$ and $f_Y(y)$ respectively, then their joint distribution has PDF

$$ f_{X, Y}(x, y) = \frac{f_X(x) f_Y(y)}{\int_{x=0}^\infty \int_{y=0}^5 f_X(x) f_Y(y) \, dy \, dx} $$

That is, the joint PDF is the product of the individual PDFs, normalized appropriately.

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