1. Three times we randomly draw a number from the following numbers:

1 2 3.

If Xi represents the ith draw, i = 1, 2, 3 then the probability mass function of Xi is given by

And P(Xi = a) = 0 for all other a. We assume that each draw is independent of the previous draws. Let X? be the average of X1, X2, and X3, i.e.,

a. Determine the probability mass function px? of X?.

b. Compute the probability that exactly two draws are equal to 1.

2. You decide to play monthly in two different lotteries, and you stop playing as soon as you win a prize in one (or both) lotteries of at least one million euros. Suppose that every time you participate in these lotteries, the probability to win one million (or more) euros is pi for one of the lotteries and 122 for the other. Let M be the number of times you participate in these lotteries until winning at least one prize. What kind of distribution does M have, and what is its parameter?

3. A box contains an unknown number N of identical bolts. In order to get an idea of the size N, we randomly mark one of the bolts from the box. Next we select at random a bolt from the box. If this is the marked bolt we stop, otherwise we return the bolt to the box, and we randomly select a second one, etc. We stop when the selected bolt is the marked one. Let X be the number of times a bolt was selected. Later (in Exercise 21.11) we will try to find an estimate of N. Here we look at the probability distribution of X.

a. What is the probability distribution of X? Specify its parameter(s)!

b. The drawback of this approach is that X can attain any of the values 1,2, 3, ... , so that if N is large we might be sampling from the box for quite a long time. We decide to sample from the box in a slightly different way: after we have randomly marked one of the bolts in the box, we select at random a bolt from the box. If this is the marked one, we stop, otherwise we randomly select a second bolt (we do not return the selected bolt). We stop when we select the marked bolt. Let Y be the number of times a bolt was selected. Show that P(Y = k) = 1/N for k = 1,2, , N (Y has a so-called discrete uniform distribution).

c. Instead of randomly marking one bolt in the box, we mark in bolts, with rn smaller than N. Next, we randomly select r bolts; Z is the number of marked bolts in the sample. Show that

(Z has a so-called hypergeometric distribution, with parameters m, N, and r.)

4. The probability distribution of a discrete random variable X is given by

100 Expectation and variance

a. Compute E[X].

b. Given the probability distribution of Y = X2 and compute E(Y) using the distribution of Y.

Transcribed Image Text:

a P(X₁ = a) 123 1 1 -102 3 col- 3 -102 3 X X1 + X2 + X3 3 P(Z = k) = N-m (7) (-) (C) for k= 0,1,2,...,r. P(X=-1)=, P(X=0)=3, P(X = 1) = ². a P(X₁ = a) 123 1 1 -102 3 col- 3 -102 3 X X1 + X2 + X3 3 P(Z = k) = N-m (7) (-) (C) for k= 0,1,2,...,r. P(X=-1)=, P(X=0)=3, P(X = 1) = ².

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1 The new transformed random variable is provided as follows The probability mass function of the random variable Xis are provided as follows Taking the probability on the both sides of transformed ra... View the full answer