Test Problems for Probability Theory ,

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1 1 Test Problems for Probability Theory , Write down the following probability density functions and compute their moment generating functions. (a) Binomial distribution with mean 30 and variance 1. (b) Poisson distribution with variance 4. (c) Exponential distribution with variance 4. (d) Normal distribution with mean 3, variance 4. (e) χ distribution with the degrees of freedom 1.. Let a r.v. X have the probability density function f(x) = π sin(πx), 0 x 1. (a) Find the mean E(X) and Var(X). (b) Find the c.d.f. F (x) = P (X x). (c) Find the 5th percentile. (d) Find the median. (e) Find the 3rd quartile. 3. A box contains five marbles numbered 1 through 5. The marbles are selected one at a time without replacement. A match occurs if marble numbered k is the kth marble selected. Let the event A i, denote a match on the ith draw, 1 i 5. (a) Find P(A i ) for 1 i 5. (b) Find P(A i A j ), where 1 i < j 5. (c) Find P(A i A j A k ), where 1 i < j < k 5. (d) Find P(A i A j A k A m ), where 1 i < j < k < m 5. (e) Find P(A 1 A A 3 A 4 A 5 ). 4. Let X have a logistic distribution with p.d.f. f(x) = e x /(1 + e x ), < x <

2 Show that Y = 1/(1 + e X ) U(0, 1) 5. Suppose that 000 points are independently and randomly selected from the unit square S = {(x, y) : 0 x, y 1}. Let Y equal the number of points that fall in R = {(x, y) : x + y 1 and x y 1}. (a) How is Y distributed? (b) Give the mean and variance of Y. (c) What is the expected value of Y /500? (d) What is P(Y 100)? 6. If the moment-generating function of X is M(t) = (1 t) 1, t < 1/. Find E(X) and Var(X). 7. Let X have the p.d.f. f(x) = θx θ 1, 0 < x < 1, 0 < θ <, and let Y = θlnx. (a) What is the moment-generating function of Y? (b) How is Y distributed? 8. Let X 1 b(n 1, p) and X b(n, p) be independent r.v. s. Define Y = X 1 + X. (a) What is M Y (t)? (b) How is Y distributed? 9. Show that the sum of n independent Poisson random variables with respective means λ 1, λ,..., λ n is Poisson with mean λ = n i=1 λ i. 10. Let Z i N(0, 1), for 1 i n and define W = n i=1 Zi. (a) Find the moment-generating function for Z 1. (b) Find the moment-generating function for W. (c) How is W distributed?

3 11. Let X = [X 1, X ] t have a multivariate normal distribution with mean vector 4 covariance matrix. 10 Write down the probability density function of X. [ 1 ] 3 and 1. For the lognormal distribution p(x) = ( ) 1 [lnx θ] exp, x > 0 πσx σ Find the maximum likelihood (ML) estimates of θ and σ for a sample of size N, respectively. 13. In no more than 100 words to explain (a) The purpose of principal component analysis (PCA). (b) The purpose of linear discriminant analysis (LDA). (c) The purpose of cluster analysis (CA). (d) What is the difference between PCA and LDA? (e) What is the difference between (PCA,LDA) and CA? Solutions for Problems Write down the following probability density functions and compute their moment generating functions. (a) f(x) = C(50, x)(0.6) x (0.4) 50 x, 0 x 50; M(t) = ( e t ) 50 (b) f(x) = e 4 4 x /x!, x = 0, 1,, ; M(t) = e 4(et 1) (c) f(x) = 1 e x/, x > 0; M(t) = 1 1 t

4 4 (d) f(x) = 1 π e (x 3) /8, < x < ; M(t) = e 3t+t (e) f(x) = 1 Γ(1/) 6 x 5 e x/, x 0; M(t) = 1 (1 t) 6 (a) E(X) = 1, V ar(x) = 1 4 π. (b) F (X) = 1 (1 cos πx), 0 x 1. (c e) x 0.5 = 1 3, median = 1, q 3 = 3 3(a) P (A i ) = 4! 5!, 1 i 5. 3(b) P (A i A j ) = 3!, where 1 i < j 5. 5! 3(c) P (A i A j A k ) =!, where 1 i < j < k 5. 5! 3(d) P (A i A j A k A m ) = 1!, where 1 i < j < k < m 5. 5! 3(e) P (A 1 A A 3 A 4 A 5 ) = 1 1! + 1 3! 1 4! + 1 5! 4. P (Y y) = P (F (X) y) = P (X F 1 (y)) = F (F 1 (y)) = y 5. This problem comes from the book we sent you earlier this semester. 5(a) Y b(000, 1 ). 5(b) E(Y ) = 1000, V ar(x) = (c) E(Y/500) = (d) P (Y 100) = k=0 ( 000 k ) ( 1 )k ( 1 )000 k.

5 5 6. E(X) = 4, V ar(x) = 48, where X χ (4). 7. (a) f(y) = 1 e y/, y > 0; (b) Exponential() 8. (a) (1 p + pe t ) n 1+n, (b) b(n 1 + n, p) 9. n e λ i(e t 1) n = e λ(et 1), where λ = λ i i=1 i=1 10. W χ (n) 11. f(x) = 1 1π exp [ 1 (x u)t C 1 (x u) ], where u = 1. [ 1 ] and C 1 = ˆ θ ML = 1 N N ln(x k ), k=1 ˆ σ ML = 1 N N (ln(x k ) k=1 θ ˆ ML )

6 6 14. Write down the following probability density functions and compute their moment generating functions. (a) Binomial distribution with mean 40 and variance 8. (b) Poisson distribution with variance. (c) Exponential distribution with mean. (d) Normal distribution with mean 5, variance. (e) χ distribution with the degrees of freedom Let a r.v. X have the probability density function f(x) = 1 sin(x), 0 x π. (a) Find the mean E(X) and Var(X). (b) Find the c.d.f. F (x) = P (X x). (c) Find the 5th percentile. (d) Find the median. (e) Find the 3rd quartile. 16. A box contains five marbles numbered 1 through 4. The marbles are selected one at a time without replacement. A match occurs if marble numbered k is the kth marble selected. Let the event A i, denote a match on the ith draw, 1 i 4. (a) Find P(A i ) for 1 i 4. (b) Find P(A i A j ), where 1 i < j 4. (c) Find P(A i A j A k ), where 1 i < j < k 4. (d) Find P(A 1 A A 3 A 4 ). 17. Let X 1, X, X 3, X 4, X 5 be a random sample of Poisson distribution with variance 5. Define Y = X j. j=1 (a) Find E(e tx 1 ). (b) Find the moment generating function of X. (c) Find the moment generating function of Y. (d) Compute E(Y ) and V ar(y ). (e) Name the distribution of Y.

7 7 Problems from ISA Entrance Exams (5%)1. Let the random variable X have the moment-generating function M(t) = e 3t+t. (a) Give the probability density function of X. (b) Find the mean and variance of X, respectively. (c) Let Y = (X 3)/, how is Y distributed? (d) Let Z = Y, how is Z distributed? (1%). Let X equal the number of bad records in each 100 feet of a used computer tape. Assume that X has a Poisson distribution with mean.5. Let W equal the number of feet before the first bad record is found. (a) Give the mean number of flaws per foot. (b) How is W distributed? (c) Give the mean and variance of W. (d) Find P(W 0) and P(W > 40), respectively. (6%)3. Let X have a geometric distribution. (a) Give the probability density function of X. (b) Show that P (X > (k + j) X > k) = P (X > j), where k, j are nonnegative integers. (6%)4. Let Y have a binomial distribution with mean 6 and variance 3. (a) Give the probability density function of Y. (b) Find P (Y ). (13%)5. Let W have a Poisson distribution with variance 3. (a) Give the probability density function of W. (b) Find the moment-generating function M W (t). (c) Find P (W ).

8 8 (10%)6. Let W 1 < W < < W n be the order statistics of a random sample of zise n from the uniform distribution U(0,1). (a) Find the probability density function of W 1. (b) Find the probability density function of W n. (c) Use the result of (a) to find E(W 1 ). (d) Use the result of (b) to find E(W n ). (7%)7. A random sample of size 16 from the normal distribution with mean µ and variance 5 yielded the estimated µ = Find a 95% confidence interval for µ. (8%)8. The length in centimeters of n = 9 fish yielded an average length of x = 16.8 and σ = Determine the size of a new sample so that [ x 0.5, x + 0.5] is an approximate 95% confidence interval for the mean.