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# Probability density function

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) Let X and Y have joint probability density function f(x,y) (s,t) = ce ^ -(s + 2t) for
0 <= s, and 0 <= t. Find
(a) c
(b) Pr {min (X, Y) 1/3}
(c) Pr {X <= Y}
(d) The marginal probability density function of X
(e) E [XY]

5) Let X and Y be independent uniform (0,1) random variables. Compute
(a) Pr {X < Y}
(b) Pr {X = Y}
(c) The probability density function of X + Y
(d) Var[X]
(e) Var[X + Y]

##### Solution Summary

Given a joint density function, this shows how to compute a marginal probability density function; given the type of variable, this shows how to compute probability density function.

##### Solution Preview

1)

a) We must have:

Therefore:

c=2.

b) I think that must be Pr{min(X, Y)<=1/3}. Then:

c) Pr{X<=Y}:
There are two methods for solving this problem.
Method 1:

Method 2:

remember that s represents X and t represents Y.

d) The ...

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