Paper 2, Section II, F

Probability | Part IA, 2007

Let XX and YY be independent non-negative random variables, with densities ff and gg respectively. Find the joint density of U=XU=X and V=X+aYV=X+a Y, where aa is a positive constant.

Let XX and YY be independent and exponentially distributed random variables, each with density

f(x)=λeλx,x0f(x)=\lambda e^{-\lambda x}, \quad x \geqslant 0

Find the density of X+12YX+\frac{1}{2} Y. Is it the same as the density of the random variable max(X,Y)?\max (X, Y) ?

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