5) The joint probability function of two discrete random...

5) The joint probability function of two discrete random variables X and Y is given by f(x,y)=cxy for x=1,2,3 and y=1,2,3, and equals zero otherwise. Find a) The constant c b) P(X=2, Y=3) c) P(1 ? X ? 2, Y ? 2) d) P(X ? 2) e) P(Y = 3) 6) Find the marginal probability function of random variables X and Y in question (5) above. Determine whether X and Y are independent. 7) For the distribution of question (5), find the conditional probability function of: a) X given Y b) Y given X

Transcript

00:01 So in this question we're told that the joint pmf for x and y is f of x and y is c xy for x equals 1 2 3 and y equals 1 2 3 and 0 otherwise.

00:21 So first of all we want to find the constant c and that means that the sum of x and y of f of x and y should be equal to 1 because it's a probability.

00:30 So 1 is equal to c times the sum from x equals 1 to 3 of x times the sum from y equals 1 to 3 of x times the sum from y equals 1 to 3 of sorry no that's not right x y and then we want a sum from y equals 1 to 3 of that as well.

01:05 So the sum of x from x from y equals 1 to 3 of that as well.

01:07 To 3 is going to be, so we've got c, the sum from 1 to 3, 1 plus 2 plus 3 is 6.

01:13 So sum from y equals 1 to 3 of 6 y.

01:17 But we know the sum from 1 to 3 is 6.

01:21 So this gives us 36 c.

01:24 So c is 1 over 36.

01:28 So we have f of x and y is 1 over 36 x y, x y equals 1, 2, 3.

01:39 So now, probability that x equals 2 and y equals 3 is 1 over 36 2 times 3, 6 over 36 or 1 6th.

01:54 The probability that 1 less than x less than or equal to 2, y less than equal to 2 is going to be the sum from x equals 1 to 2, of 1 over 36 xy.

02:16 So this is going to be 1 over 36 times the sum from x equals 1 to 2.

02:21 And then 1 plus 2 is 3, so 3x.

02:26 So this is going to be 9 over 36.

02:29 And if we divide by 3, we can get 3 over 12, and we can divide by 3 again to get a quarter.

02:39 So now part d, probability that x is greater than equal to 2 is going to be the sum from y equals 1 to 3, sum from x equals 2 to 3, 1 over 36 xy...

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