4.1. Suppose X and Y are independent random variables, which are jointly continuous. 1. Show carefully that the distribution of 'X + Y given X = x' is equal to the distribution of the random variable x +Y. Hint: Consider the change of variables h(x, y) = (x,x+y). 2. Suppose X and Y each have an exponential distribution with parameter a, find the joint p.d.f of X and X + Y. 3. In the situation of (ii) above calculate E[X | X + Y].

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4.1. Suppose X and Y are independent random variables, which are jointly continuous.
x' is equal to the distribution
1. Show carefully that the distribution of 'X + Y given X
of the random variable à +Y.
Hint: Consider the change of variables h(x, y) = (x,x+y).
=
2. Suppose X and Y each have an exponential distribution with parameter a, find the joint
p.d.f of X and X + Y.
3. In the situation of (ii) above calculate E[X | X + Y].
Transcribed Image Text:4.1. Suppose X and Y are independent random variables, which are jointly continuous. x' is equal to the distribution 1. Show carefully that the distribution of 'X + Y given X of the random variable à +Y. Hint: Consider the change of variables h(x, y) = (x,x+y). = 2. Suppose X and Y each have an exponential distribution with parameter a, find the joint p.d.f of X and X + Y. 3. In the situation of (ii) above calculate E[X | X + Y].
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