srikarmargam

Suppose that Y is a random variable with the probability mass function, 2 k PſY = k] = nom 1, for k=0, 1, ..., n - 1, n (n − 1)? where n > 2. 1. Derive the expected value of Y. 2. Evaluate the second moment of Y. 3. Determine the variance of Y.

Suppose that X is a discrete random variable with the probability mass function, 2 (n – k). P [X = k] = for k= 1, 2, ..., n, n(n − 1) 1 where n > 2. 1. Derive the expected value of X. 2. Determine the variance of X. 3. Derive the second moment of X.

Problem 3. (30%) Sec. 11.1 #3 Let X be a discrete random variable with the probability mass function p(i) = 2)", for i = 1,2,3,... zero elsewhere. (a) (10%) Find Mx(). (b) (10%) Using Mx(), find E[X]. (c) (10%) Using Mx(t), find Var[X]. Solution: Type your answer here.

A discrete random variable X has the following probability mass function: P(X = x) = w Sk(1 – x)2 x = -1,0,1,2 otherwise (a) Show that k = 1 Find E(X) (e) Show that E(X2) = (d) Find var(1 – 3X)

(b) The joint probability mass function for a pair of discrete (integer-valued) random variables X and Y is Px,y(n, m) = 60 ), nm > 0, 70. no () Find the marginal mass functions of X and Y. Pyin) (ii) Are X and Y independent? Justify your answer. (iii) Show that the probability mass function for X + Y is Otherwise. y Px(n) = < 3 (3), 720 Sinzo ), náo. [4] (ii) Show and y indepens functions Px+v(n) = { (3) - n20, otherwise.

3. A fair die is tossed independently and repeatedly. Let X be the number of tosses needed to turn up the first 3. (a) Find P(X = 1), P(X = 2) and P(X = 3). (b) Find the probability mass function of X. (c) Show that the probability mass function you found in part (b) is a valid probability mass function

1. a) A biased coin has probability p of landing heads. Show that the num- ber of tosses of this coin, X, necessary to obtain the first head has probability function P[X = x] = p(1 – p)2=1 for x = 1, 2, ... b) A coin was tossed and the number of tosses needed to obtain the first head was recorded. This was repeated 200 times and the numbers are given in the following table. Number of tosses 1 2 3 4 5 6 Observed frequency 116 54 16 6 4 4 Carry out a goodness of fit test to examine the hypothesis that the coin is a fair one, that is it has probability 0.5 of landing heads. Find the P value and explain your conclusion.

Solve for z.

Four coins are tossed, and the number of heads is recorded. Shade the region on the histogram that gives the indicated probability. P(x < 2) || Which of the following shaded regions gives the indicated probability? (Type A, B, C, or D.) 0.57 0.387 0.257 0.137 A) 001234 0.5) 0.38 0.257 0.137 B) 001237 0.51 0.5) 0.38 0.38 0.25 0.13 0 IL 0.257 0.131 L OD 0 1 2 3 4 0 1 2 3 4

Problem 4 Random variables X and Y can take values from the set {0, 1, 2). Samples of size 10 give the results recorded in the table. 0 1 2 2060 20 21 44 35 Now suppose that the table was produced by the following experiment. There are two identical (biased) coins with probability of head equal p, and X is the number of heads if both coins are tossed. Also there are two other identical (biased) coins with probability of head equal p, and Y is the number of heads if both coins are tossed. Given the table at 5% significance level test null hypothesis of P = pagainst the alternative pop

let X be the number of heads in three of a coin whole Probability of heads is p. Suppose Ho ip= Ž H. iD=3 let Q(x) be defined so that its resects Ho in favor of Hi ię x 720 Find the size of this test and then evaluate its power.

What is the probability of randomly selecting a score less than X = 52?

P(X < 52) =

 

What is the probability of selecting a random sample of n = 9 scores with a mean less than M = 52?

P(M < 52) =

(Look at photo)

 

What is the probability of selecting a random sample of n = 16 scores with a mean less than M = 52?

P(M < 52) =