Notes on probability : Exercise problems, sections (1-7)

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1 Notes on probability : Exercise problems, sections (1-7) 1 Random variables 1.1 A coin is tossed until for the first time the same result appears twice in succession. To every possible outcome requiring n tosses attribute probability l /2 n-1. Define an appropriate* random variable, ( i.e., give Ω X, P X ) for this experiment. 1.2 Give an example of two equivalent random variables (that is, two random variables having the same distribution) that are defined on the same space but are never equal. 1.3 From a population of n distinct elements a sample of size r is taken, (a) without, (b) with replacement. Define appropriate random variables for these experiments. 1.4 A cell contains N chromosomes, between any two of which an interchange of parts may occur. Totally r random interchanges are known to occur. Each interchange can occur between any 2 chromosomes. Define an appropriate random variable to study this phenomenon. 1.5 A class consisting of 4 graduate and 12 undergraduate students is randomly divided into 4 groups of 4. Define an appropriate random variable. *( By appropriate we mean that the outcome of the experiment can be known by knowing the value of the random variable) 2 Events 2.1 In 1.1, Find the probability of the following events: (a) the experiment ends before the sixth toss, (b) an even number of tosses is required. 2.2 Two dice are thrown. Let A be the event that the sum of the faces is odd, B the event of at least one 6. Describe the events AB, A U B, AB c. Find their probabilities assuming that all 36 sample points have equal probabilities. 2.3 Given thirty people, find the probability that among the twelve months there are six containing two birthdays and six containing three.

2 2.4 We place at random n particles in m distinct boxes, with m>n. Find the probability p that the particles will be found in n particular boxes (with one ball in each box). Consider the following cases : a) Maxwell - Boltzmann distribution- the particles are distinct. All alternatives are possible. b) Bose - Einstein distribution - the partices cannot be distinguished. All alternatives are possible. c) Fermi - Dirac distribution - the particles cannot be distinguished. At most one particle in a box. 2.5 From a population of n elements a sample of size r is taken. Find the probability that none of N prescribed elements will be included in the sample, assuming the sampling to be (a) without, (b) with replacement. Compare the numerical values for the two methods when (i) n = 100, r = N = 3, and (ii) n = 100, r = N = Negative binomial distribution: Consider a sequence of independent experiments where the probability of success of each experiment is p, 0<p<1, and q=1-p. We terminate the experiment with the k th success. Let X be the number of failures prior to the k th success. Show that P(X=x) = k+x-1 C x p k q x, x= 0,1, In 1.4, what is the probability exactly m chromosomes will be involved? 3 Axioms of probability 3.1 Prove Boole s inequality (section 3.1 in notes) 3.2 Prove the continuity theorems. 3.3 In discrete case, do the first three axioms together imply the fourth axiom? Prove or provide counter-example. 4 Cumulative Distribution Funcion 4.1 Prove the properties of CDF. 4.2 For a non-negative random variable X with CDF F(.), Show that F c (x)dx = EX, where the integration is from 0 to. How will you extend this for a general random variable?

3 5 Independent Events 5.1 Consider an experiment involving two successive rolls of a 4-sided die in which all 16 possible outcomes are equally likely and have probability 1/16. (a) Are the events Ai = {1st roll results in i}, Bj = {2nd roll results in j}, independent? (b) Are the events A = {1st roll is a 1}, B = {sum of the two rolls is a 5}, independent? (c) Are the events A = {maximum of the two rolls is 2}, B = {minimum of the two rolls is 2}, independent? 5.2 Is the equality P(A 1 A 2 A 3 ) = P(A 1 ) P(A 2 ) P(A 3 ) enough for independence. Prove or provide counter-example. 5.3 Let A and B be independent events. Show that A and B c are independent. Show also that A c an B c are independent. 5.4 Suppose a coin is tossed twice where the probability of a head is p on the first toss and q on the second. Assume that the outcomes of different tosses are independent. Let X be the number of heads that occur on the two tosses. a) Find the probability of each of the events {X=0},{X=1},and {X=2}. b) Show that P(X=0) = P(X=2) if and only if p+q = 1. c) Use (b) to show that if the tosses are independent, the three events in (a) can never be equally likely. 5.5 If events A and B are independent, does it always mean that the occurrence of A tells us nothing about the occurrence of B? 5.6 An internet service provider has installed c modems to serve the needs of a population of n customers. It is estimated that at a given time, each customer will need a connection with probability p, independently of the others. What is the probability that there are more customers needing a connection than there are modems? 6 Independent Random Variables 6.1 Let X and Y be two random variables with joint pmf p XY (k, j) = C*K/(j + 1); j = 1,,N; k = 1, 2,, j. (a) Find C. (b) Find p Y (j). (c) Are X and Y independent?

4 6.2 Show that k random variables are mutually independent if and only if a) their joint pmf is the product of the marginal pmf s. b) their joint cdf is the product of the marginal cdf s. 6.3 Suppose that X and Y are independent binomial random variables with parameters (n,p) and (m,p). Argue probabilistically (no computations necessary) that X +Y is binomial with parameters (n+m,p). What if X and Y are not independent? 6.4 For any k > 0, define k random variables that are k-1 wise independent, but not mutually independent. 6.5 Let X and Y be independent random variables with geometric distribution, with parameter p. Find (a) P{X = Y} (b) P{X >= 2Y} 7 Elemantary Conditional Probability 7.1 On Sunday September 9, 1990, the following question appeared in the Ask Marilyn column in Parade, a Sunday supplement to many newspapers across the United States: Suppose you re on a game show, and you re given the choice of three doors; behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what s behind the doors,b opens another door, say No. 3, which has a goat. He then says to you, Do you want to pick door No. 2? Is it to your advantage to switch your choice? Craig F. Whitaker, Columbia, Md 7.2 Let X and Y be independent geometric random variables having the same parameter p. Guess the value of P{X = i X +Y = n} Hint: Suppose a coin having probability p of coming up heads is continually flipped. If the second head occurs on flip number n, what is the conditional probability that the first head was on flip number i, i = 1,..., n 1? Verify your guess analytically. 7.3 If X and Y are independent binomial random variables with respective parameters (n1,p) and (n2,p), calculate the conditional probability mass function of X given that X + Y = m. The distribution is known as the hypergeometric distribution. 7.4 The Mongol general Subudai is expecting reinforcements from Chenggis Kahn before attacking King Bela of Hungary. The probability mass function describing the number N of tumens (units of 10,000 men) that he will receive is P N (k) = cp k ; k = 0, 1, If he receives N = k tumens, then his probability of losing the battle will be 2 -k. This can be described by defining the random variable W which will be 1 if the battle is won, 0 if the battle is lost, and defining the conditional probability mass function P W/N (m k) = Pr(W = m N = k) = { 2 -k m = k m = 1. (a) Find c.

5 (b) Find the (unconditional) pmf PW(m), that is, what is the probability that Subudai will win or lose? (c) Suppose that Subudai is informed that definitely N <10. What is the new (conditional) pmf for N? (That is, find Pr(N =k N <10).) 7.5 Chain rule: Show that P(A n,a n-1 A 1 ) = P(A n A n-1, A 1 ) P(A 2 A 1 ) P(A 1 ) 7.6 Let X 1,..., X n be independent random variables having a common distribution function that is specified up to an unknown parameter θ. Let T = T (X) be a function of the data X = (X 1,..., X n ). If the conditional distribution of X 1,..., X n given T (X) does not depend on θ then T (X) is said to be a sufficient statistic for θ. In the following cases, show that T (X) = X i is a sufficient statistic for θ. (a) The mass function of X i is p(x) = θ x (1 θ) 1 x, x = 0, 1, 0 < θ <1. (b) The X i are Poisson random variables with mean θ. References 1.1 Feller1, 3 rd ed,i Feller1, 3 rd ed,ii Feller1, 3 rd ed,iv MIT lecture notes(2000) eg Feller1, 3 rd ed,i Feller1,3 rd ed,ii Papoulis,3 rd ed, Feller1, 3 rd ed,ii Wolff, Feller1, 3 rd ed,iv MIT lecture notes(2000), eg MIT lecture notes(2000), eg Wolff, Wolff, Wolff, MIT lecture notes(2000), eg Gray & Davisson, Ross, 9 th ed, Feller1,3 rd ed,ix A Modern Introduction to probability and Statistics(MIPS)- F.M. Dekking C., Kraaikamp,H.P. Lopuhaa, L.E. Meester, Ross, 9 th ed, Ross, eg Gray & Davisson, Papoulis,3 rd ed, Ross, 9 th ed, 3.18

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