Problem 4 (30 points) The joint PMF of the random variables...

Problem 4 (30 points) The joint PMF of the random variables X and Y is given by the following table: egin{tabular}{|c|c|c|c|} hline y & x = 1 & x = 2 & x = 3 \ hline y = 3 & c & c & 2c \ y = 2 & 2c & 0 & 4c \ y = 1 & 3c & c & 6c \ hline end{tabular} • (a) Find the value of the constant c. • (b) Find $p_Y(2)$. • (c) Consider the random variable $Z = YX^2$. Find $E[Z mid Y = 2]$. • (d) Conditioned on the event that $X eq 2$, are X and Y independent? Give a one-line justification. • (e) Find the conditional variance of Y given that X = 2.

Transcript

00:01 I'm david and i'm here to have you on change the question.

00:02 In the question here we're given the term vmf on the two random variables x and y.

00:08 Now let me bring it up here.

00:10 And from here in the first question a, ask me to find the volume the c.

00:15 To find the c we have to add up on the probability and then it must exactly equal to 1.

00:20 So we have the total here, it will equal to the 1.

00:24 And if i add up everything we have 1 plus 1 plus 2, plus 2, plus 2 plus 4 plus 3 plus 1 plus 6 and it equal to the 20 c therefore the c will equal to 1 over the 20 and you get equal to the 0 05 and therefore we can formally a new table here to make it nice and we have this will be the x and the y so this will be the y this will be the x and the x.

01:01 1 can take the value of the 1, 2, 3, i will order them in the nice value will be 1, 2 and 3.

01:12 And we will have now this will be the total, and then i can find the marginal on the x and also the marginal of the y as well.

01:26 Now the inside we will have found the first one will be 1 -1.

01:31 Again, the 3c.

01:32 So will be the 0505 times the 3 equal to 0 .115.

01:38 X, equal to the 1, y, equal to the 2, x to the 1, y equal to the 2.

01:47 It will be the 2c, will be 0 .1 1.

01:51 And this one will be c only, it will be the 0 .05.

01:55 The next one, we will have x x2, y, equal to the 1, will be the c only.

02:01 And this one will be the 0 .0 .0 .05.

02:06 And this one will be the 6c will be the 0 .3.

02:10 This one 4c will be the 0 .2.

02:13 And it will be the 0 .1.

02:15 Now, if you find the marginal, we'll add up everything to get the probability x equal to the 1.

02:21 So it will add everything we get the 0 .2 plus 0 .1 .1 will be the 0 .3.

02:25 There will be the 0 .1.

02:27 And this will be the 0 .6.

02:29 Here we have this one will be the 0 upon 0 .2 plus 7 .3 will be 0 .5.

02:37 This will be the 0 .3 and this will be the 0 .2.

02:42 And that will be the summary of the question a.

02:45 Now we will be able to find the answer for the question b...

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