i need their solutions please ekeetpxkte 01 the following result could be used without proofif x

1a. To check that T follows the geometric distribution Gp on N, we need to show that PT = k = p(

I need their solutions please ekeetpxkte 01 the following...

E[] = keEtPX = k, te [0,1]. The following result could be used without proof: if X and Y are two independent random variables E-valued, then 9x + yt = gxtgytVt [0,1]. An urn contains a proportion p of white balls and q black balls, where p + q = 1. At each instant n > 1, we choose a ball from the urn and immediately replace it. We call T the r.v. counting the number of necessary selections until we obtain the first white ball. 1a. Check that T follows the geometric distribution Gp on N, that is PT = k = p(1-p)^(k-1) for k âˆˆ N. 1b. Express the generating function, then the expectation of T. 1c. Let X be a sequence of i.i.d random variables having the same distribution as T. Compute the generating function of S = X + X + ... + X (n times). 2. We call T' the r.v. counting the number of necessary selections until we obtain exactly r white balls. 2a. Show that T' follows the Pascal distribution Pr,p), given by PT' = k = C(p+q-1, r-1) * p^r * q^(k-r). 2b. Deduce the generating function of T. 3. Compare the distribution of S defined in 1c to the one of T.

Transcript

00:01 So in this question, we draw, so we have an urn, which contains seven white and three black balls.

00:11 And first, we take two balls from the urn, and x is the number of white balls.

00:23 And then, without replacement, we take another one ball, and y is the number of white balls.

00:38 So first of all, we want to compute the joint distribution of x and y.

00:44 So the probability that x equals 0 is going to be the probability of choosing two blackballs, which is going to be 3 tenths times 2 9ths, which is 6 over 90, which is 1 over 15.

01:04 The probability that x is equal to 1 is going to be, so you can either choose it first or second, so there are 2nd.

01:12 Two ways round of doing it.

01:15 And then it'll be three over ten to choose a black ball first, and then seven over nine to choose the white ball second.

01:30 But if it was the other way round, it would be seven over ten, and then three over nine.

01:34 And since you're multiplying them together, we can just do it this way.

01:39 So this gives us six times seven over 90.

01:43 But six over 90 is one over 15 now.

01:48 And the probability that x equals 2, well it's the only other possibility, which means it must be the complement of these two.

01:55 So that's going to be 1 minus 8 over 15, which is 7 over 15.

02:01 Now, y can either be 0 or 1.

02:06 So the probability that y equals 0, given that x equals a little x, is going to be, so if you take out x white balls, then there's going to be 7.

02:20 Minus x white balls remaining.

02:23 So the probability of picking out a black ball is going to be 3 over 8, in this case.

02:34 The probability that y equals 1 given that x is equal to x.

02:38 Sorry, so this is going to be, so if we pick out x white balls, that means we pick out two minus x black balls.

02:50 So the probability of picking out a blackball is going to be 3 minus 2 minus x.

02:58 So that's 1 plus x over 8.

03:09 And the probability that y equals 1 given x equals x is just going to be the complement of this.

03:14 So that's 7 minus x over 8.

03:20 So now we can write down the joint distribution.

03:24 So if we have y going down this way 01, x going over here, 012, and then the joint distribution, property x equals x, and y equals y is going in this way.

03:40 Let's see what this is going to be.

03:47 So y equals 0 given x equals 0 is 1 8th.

03:52 The probability that x equals 0 is 115th.

03:56 So we multiply those together to get 1 over 120.

04:02 The probability that y equals 0 given x equals 1 is 1 quarter.

04:06 Probability that x equals 1 is 7 over 15, so we get 7 over 60.

04:14 Y equals 0 given x equals 2 is 3 8th, and 3 8th times 7 over 15 is going to be 1 8th times 7 over 5, which is 7 over 8 times 5, 7 over 40.

04:30 And then y equals 1.

04:35 So we're going to have, so if x equals 0, we've got 7.

04:38 And we need to multiply that by one -fifteenth.

04:42 So that's 7 over 8 times 15, which is 7 over 120.

04:49 Y equals 1, x equals 1.