Suppose the random variables x y have the joint probability...

Suppose the random variables X, Y have the joint probability density function f(x,y) = ce-2x-3y for 0 < x and x < y. Determine the following: i) E(Y | X =1) ii) P(Y<2|X=1) iii) Conditional probability distribution of X given Y = 2

Transcript

00:02 Given the joint probability density function of x, y is f of x, y equals c into e to the power minus 2x minus 3y for x is greater than 0 and x is less than y and 0.

00:21 Otherwise, first we will find the c value.

00:27 For this we know the total probability that is x, y f of x, y dy dx equals 1.

00:36 So now x limits are from 0 to infinity and y is from x to infinity f of x, y is c into e to the power minus 2x minus 3y dy dx equals 1.

00:49 Now this gives 0 to infinity c e to the power minus 2x minus 3y upon minus 3 limits x to infinity dx equals 1.

01:01 This gives minus c by 3 0 to infinity e to the power minus 2x minus 3x by lower limit and infinity 0 dx equals 1.

01:15 Therefore this is c by 3 0 to infinity e minus 5x dx equals 1.

01:23 This gives c by 3 e to the power minus 5x upon minus 5 0 to infinity equals 1.

01:32 This gives c upon minus 15 minus of e to the power minus 5 into 0 equals 1 gives c upon 15 equals 1 implies that c.

01:43 Now we will find marginal function of x that is f of x equals y limit f of x, y that is joint period dy.

01:59 Now y limits are from x to infinity and joint is 15 e minus 2x minus 3y dy.

02:08 This gives equals 15 e to the power minus 2x minus 3y upon minus 3 limits of infinity.

02:20 This gives minus 5 e to the power minus 2x minus 3x this will be minus since it is applying lower limit...

Question

Suppose the random variables X, Y have the joint probability density function f(x,y) = ce-2x-3y for 0 < x and x < y. Determine the following: i) E(Y | X =1) ii) P(Y<2|X=1) iii) Conditional probability distribution of X given Y = 2

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