550.3 Probability & Statistics - FALL 008 FINAL EXAM NAME. An urn contains white marbles and 8 red marbles. A marble is drawn at random from the urn 00 times with replacement. Which of the following is closest to the probability of drawing more than 75 red marbles? (a). (b).6 (c).84 (d).87 (e).95. A system has two components placed in series so that the system fails if either of the two components fails. The second component is twice as likely to fail as the first. If the two components operate independently and if the probability that the entire system will fail is.8 then what is the probability the first component will fail? (a).0 (b).0 (c).4 (d).8 (e).56 3 3 3. Let Z Z Z 3 be independent normal random variables each with mean 0 and variance. Which of the following has a chi-square distribution with degree of freedom? (a) Z +Z (b) (Z +Z ) Z 3 (c) Z +Z Z 3 (d) (Z +Z Z 3 ) 3 (e) (Z +Z Z 3 ) 4. A pair of dice is tossed 0 times in succession. What is the probability of observing no 7 s and no s in any of the 0 tosses? (a) ( ) 8 0 (b) ( ) 30 0 ( 34 ) 0 (c) [ ( 6 )] 0 )( (d) ( [ 8 0 ) (e) ( ) ][ 6 0 ( ) ] 0 5. Let X...X m and Y...Y n be independent random samples from a normal distribution with unknown mean μ and unknown variance σ > 0. Let X = m m X j j= Y = n n j= Y j S X = m m j= (X j X) and T = c Y X S X where c is a constant. If T has a Student s t-distribution with appropriate degrees of freedom what is the value of c? (a) ( ( ) mn / / m+n) (b) (c) (m )n (d) ( ) ( ) m / / mn (e) m+n m (m+n)(m ) 6. Let Y have a uniform distribution on the interval (0 )and let the conditional distribution of X given Y = y be uniform on (0 y). What is the marginal density function of X for0<x<? (a) ( x) (b) x (c) ( x) (d) x (e) x 7. Let X and Y be independent normal random variables with means μ X =3andμ Y =5and variances σx =9andσ Y = 6. Which of the following is closest to the probability that Y X is greater than 7? (a).03 (b).6 (c).4 (d).84 (e).97 8. P (A B) =. P (A) =.6 and P (B) =.5. Then P (A B )= (a). (b).3 (c).7 (d).8 (e).9
9. Let X be a Poisson random variable with mean λ. IfP (X = X ) =.8 then what is the value of λ? (a) 4 (b) ln(.) (c).8 (d).5 (e) ln(.8) 0. Let X X and X 3 be independent random variables each having the density function f(x) = 3x for 0 <x<. Let Y =maxx X X 3 }.WhatisP(Y > )? (a) (b) 37 (c) 343 (d) 7 (e) 5 64 64 5 8 5. Let the random variable X have the moment generating function M(t) = e3t for <t<. t What are the mean and variance of X respectively? (a) and (b) and 3 (c) 3 and (d) 3 and 3 (e) 3 and 6. Let X X X 3 X 4 be a random sample from the discrete distribution θ x e θ P (X = x) = for x =0... x! where θ>0. If the data are 7 0 3 and 5 what is the maximum likelihood estimate of θ? (a) 4 (b) 8 (c) 6 (d) 3 (e) 64 3. Let X and Y have the joint density function f(x y) = x + y for 0 <x< 0 <y< What is the conditional mean E(Y X = )? 3 (a) +6y (b) (c) 5 5 3 (d) (e) 3 5 4. Let X X...X n be a random sample from a normal distribution with variance σ = 0. If P ( n j= (X j X) 44) =.05 what is the value of n? (a) 0 (b) (c) (d) 3 (e) 5 5. Let X and Y have the joint probability mass function 6 4x 4y+xy for x = 3; y = 3 p(x y) = Which of the following statements is true? (a) X and Y are dependent random variables with different marginal distributions. (b) X and Y are dependent random variables with the same marginal distributions. (c) X and Y are independent random variables with different marginal distributions. (d) X and Y are independent random variables with the same marginal distributions. (e) There is insufficient information to determine if X and Y are dependent or independent.
6. Let X...X 6 and Y...Y 8 be independent random samples from a normal distribution with mean 0 and variance and let W = 4 3 6 j= X j 8 j= Y. j What is the 99th percentile of the distribution for W? (a) 6.37 (b) 7.46 (c) 8.0 (d) 6.8 (e) 0.09 7. Let X and Y be independent random variables with E(X) = and E(Y ) = and V ar(x) = Var(Y )=σ. For what value of k is k(x Y )+Y an unbiased estimator for σ? (a) (b) (c) 3 4 (d) 4 3 (e) 8. Let X have the density function f(x) = x k for 0 <x<k For what value of k is the variance of X equal to? (a) (b) 6 (c) 9 (d) 8 (e) 9. Let X...X 9 be a random sample from a normal distribution with mean μ and variance σ. Which of the following are the endpoints for a 90% confidence interval for μ? 9 (a) x ± 0.33 j= (x 9 j x) (b) x ± 0.73 j= (x 9 j x) (c) x ± 0.305 j= (x j x) (d) x ± 0.30 9 j= (x j x) (e) x ± 0.385 9 j= (x j x) 0. Let (X Y ) be the coordinates of a point randomly chosen in the xy-plane and let R = X + Y be the distance from (X Y ) to the origin. If X and Y are independent normally distributed random variables each with mean 0 and variance σ what is the value of r such that the probability the R exceeds r is.95? (a) 0.σ (b) 0.σ (c).7σ (d).45σ (e) 5.99σ. Let X X...X n be a random sample from a distribution with density function λ e λ x μ < x<. Using the Neyman-Pearson theorem determine a critical region to test the null hypothesis H o : μ =0λ= 3 against the alternative hypothesis H a : μ =λ=3. (a) x i c (b) x i c (c) x i x i c (d) x i x i c (e) x i c. After a certain time the weight W of crystals formed is given approximately by W = e X where X is distributed normally with mean μ and variance σ > 0. What is the density function of W for 0 <w<? (a) π ln(w) e (ln(w) μ) σ (b) πσ e (w μ) σ (c) πσw e (ln(w) μ) σ (d) π ln(w) e (ln(w) μ) (ln σ) (e) πσ e (w μ) σ
3. Three people XY and Z in order roll a single die. The first one to roll an even number wins and the game is ended. The probability that X will win is (a) (b) 3 5 (c) 4 7 (d) 3 (e) 3 4 4. Let X be the random variable that counts the number tosses of a first coin needed to produce the first head. Compute E(X X ). (a) (b) p (c) +p (d) (e) + p p p p p 5. A random sample X...X n is taken from a distribution with density function f(x) =(θ+)x θ for 0 <x< and f(x) = 0 for other values of x. Assume θ>0. What is the method of moments estimator for θ? (a) X X (b) X X+ (c) X (d) n n j= ln(x j) 6. Suppose X and Y are continuous random variables with the joint density function (e) ln(x) f(x y) = 6xy for 0 <x< y<. What is the density function g(z) for the random variable Z = XY on the domain where it is positive? (a) 6z ln(z) (b) 4z ln(z) (c) 3z ln(z) (d) z (e) 3z 7. Let X X... be an independent sequence from a distribution with density function f(x) = e x for x>0; and f(x) =0forx 0. Set V n = e n minx...x n}. lim n P (V n x) for appropriate values of x is (a) e x (b) e /x (c) x (d) x (e) e x 8. If X has a normal distribution with mean 0 and variance and Y = e X then then kth moment of Y is (a) k (b) e k (c) e k (d) e k (e) k 9. Identical twins come from the same egg and hence are of he same sex. Fraternal twin have a 50-50 chance of being the same sex. Among twins the probability of a fraternal set is p and an identical set is q = p. If the next set of twins are of the same sex what is the probability they are identical? (a) q p (b) q (c) (d) q (e) q +p +q 30. Suppose X X...X n is a random sample of size n from a normal distribution with mean μ and variance σ > 0. What is the variance of S = n n j= (X j X)? (a) σ n (b) σ n (n ) (c) σ4 n (n ) (d) σ4 n 3. Let X and Y be independent random variables with respective densities: (e) σ4 n f X (x) = for 0 <x< and f Y (y) = y for 0 <y< What is P (Y <X)? (a) (b) 3 (c) 4 (d) 5 (e) 6
3. Let X be a random variable with probability density function f(x) =4x 3 for 0 <x<; and f(x) = 0 for all other values of x. What is the value of E( X )? (a) 3 4 (b) 4 3 (c) 4 5 (d) 5 4 (e) 33. A person rolls two identical-looking fair six-sided dice. As the person rolls them you notice a six came up one of the die but are unsure of the other die. The person quickly covers up both dice. What is the probability that this roll produced a sum of? (a) (b) (c) (d) (e) 6 34. If the density function of X is f(x) = for <x<; and f(x) = 0 elsewhere what is the probability density function g(x) ofy = X? 3 (a) g(x) = y for 0 <y< (b) g(x) = for 0 <y< y 4 (c) g(x) = for 0 <y< y 3y for 0 <y< (d) g(x) = y for 0 <y< (e) g(x) = Questions 35 and refer to two discrete random variables X and Y which take on values 0 with the following probability distribution: 35. What is E(Y X = )? (a) 3 (b) 6 (c) 7 (d) 9 (e) 3. What is the covariance of X and Y? (a) 4 (b) (c) (d) (e) 4 8 6 3 6 8 37. A random sample of size 4 was drawn from a normal population and the following values were observed: 7535. What is a 95% confidence interval for the mean of the population based on these experimental values? (a) (4. 5.88) (b) (7.06.94) (c) (.76.76) (d) (.454 9.546) (e) (.67 8.38) 38. A fair coin is tossed. If a head occurs fair six-sided die is rolled; if a tail occurs fair six-sided dice are rolled. Let Y be the total on the die or dice. What is E(Y )? (a) 7 (b) 5 (c) 4 (d) 7 (e) 39. Let X and Y have the joint density f(x y) =8xy for 0 <x<y<; and f(x y) =0otherwise. What is P (Y X )? (a) (b) 7 (c) 6 (d) 3 8 (e) 3 40. How do you spell the instructor s last name? (a) FLACCO (b) SMITH (c) TORCASO (d) JONES (e) ROETHLISBERGER