# 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]

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##### 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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