Reference no: EM132371877

Probability Models Assignment - Problem Set

1.A. Let X₁ be a geometric random variable with parameter p₁, and let X₂ be a geometric random variable with parameter p₂. Evaluate Pr{X₁ ≤ X₂}. (Suggestion: Condition on X₁.)

1.B. An urn contains b blue balls and r red balls. Balls are removed (without replacement) at random until the first blue ball is drawn. Show that the expected number of balls drawn is (b + r + 1)/(b + 1).

(Hint: Suppose the red balls are numbered 1, . . . . , r. What is the probability that ball number 1 is drawn before the first blue ball?)

1.C. Let X be a random variable taking values in {0, 1, 2, . . . }, and let φ_X(s) be its PGF. Let

Q(s) = _n=0Σ^∞sⁿPr(X ≤ n)

be the "generating function" of the sequence Pr(X ≤ n). Show that

Q(s) = φ_X(s)/1-s.

1.D. A random variable X (taking values in {0, 1, 2, . . .}) has the following probability generating function (PGF):

φ_X(s) = (7 - 3s)/(15 - 14s + 3s²).

(a) Find an expression for Pr(X = k) for k = 0, 1, . . .. (Suggestion: do a partial fractions expansion to express φ_X(s) as a sum of two terms of the form a/(b - cs) [many calculus texts explain how to do this bit of algebra], and then expand each term as an infinite series [write as geometric series]).

(b) Compute the first two derivatives of φ_X(s) (you may want to use the partial fractions expression), and use these to compute the mean and the variance of X.

1.E. Consider two strangely shaped dice, each with faces numbered from 1 to 6. Neither of the dice is fair (that is, the six numbers are not equally likely to occur). But we are interested in rolling the two dice and adding the numbers, getting a random result from 2 through 12. Is it possible that the probabilities for the sum is exactly the same as the probabilities that we would get from two fair dice? (More formally, do there exist independent random variables Y and Z [with different distributions], each taking values in {1, . . . , 6}, such that Y + Z has the same distribution as the sum of two fair dice?) Hint: The PGFs are polynomials; try to factor them.

ADDITIONAL PROBLEMS -

1.W. Use the result Problem 1.C to show that

_k=0Σ^∞s^kPr(X > k) = (1-φ_X(s))/(1 - s).

Observe that when you take the limit s → 1- in this equation, you re-derive an identity that we have already seen in class - what is it?

1.X. A Swedish newspaper once observed that in one year, 35 people were drowned in accidents involving small boats, and that only five of these people were wearing life jackets. The newspaper concluded that it was six times safer to wear a life jacket when boating. Use conditional probabilities to show what is wrong with this reasoning. (Notice also that if almost everybody wore life jackets, then the newspaper's logic could lead to the conclusion that it was safer NOT to wear a life jacket!)

1.Y. When a certain experiment is performed, it succeeds with probability 0.8 and fails with probability 02. It costs $10 to perform the experiment, but if it succeeds then you will earn $100. You decide to perform the experiment several times: You will stop as soon as you get one successful outcome, and you will also stop if the first three attempts all fail. Let C denote the cost of the entire operation (a profit is represented by a negative cost). Find the expected value of C.

1.Z. Let X be a random variable satisfying

Pr(X = k) = 1/(k(k + 1)) for k = 1, 2, 3, . . ..

(i) Verify that this is indeed a probability distribution (i.e., show that the probabilities add up to 1.) (Hint: Write 1/k(k + 1) as the difference of two simpler fractions. You can think of this as an example of partial fractions.)

(ii) Give a formula for the probability generating function of X in terms of the natural logarithm function.

(iii) Use two different methods to show that E(X) = ∞. (Use part (ii) and the more direct method of showing that the sum defining E(X) diverges.)

(Note: You may need to refer to the Infinite Series chapter of a calculus text when you are working on this problem.)

Textbook - "Classical and Spatial Stochastic Processes with Applications to Biology" (Second Edition) by RinaldoSchinazi; Springer E-book, 2014".

Note - Handwritten solution required.