Closed book and notes. 60 minutes. Cover page and four pages of exam. No calculators.

Transcription

1 IE 230 Seat # Closed book and notes. 60 minutes. Cover page and four pages of exam. No calculators. Score Exam #3a, Spring 2002 Schmeiser

2 Closed book and notes. 60 minutes. 1. True or false. (for each, 3 points if correct, 2 points if left blank.) (a) T F If X is normally distributed with mean µ X and standard deviation σ X, then P(X >µ X ) = P(X µ X ). (b) T F If X is normally distributed with mean µ X and standard deviation σ X, then 0.65 < P(µ X σ X <X <µ X +σ X ) <.75. (c) T F If Y is exponentially distributed with rate λ, then P(Y <0) = 0. (d) T F If approximating a binomial-distribution probability with a Poissondistribution probability, the continuity correction is useful. (e) T F If is a joint probability mass function, then (x, y ) = P(X x, Y y ) for all real numbers x and y. (f) T F If is a joint probability density function, then (x, y ) = P(X x, Y y ) for all real numbers x and y. (g) T F If X and Y are independent random variables, then P(X 6, Y 8) = f X (6) f Y (8), regardless of whether f X and f Y are pmfs or pdfs. (h) T F If (X 1, X 2,...,X 6 ) is a multinomial random vector, then X 1 + X 2 is a binomial random variable. (i) T F If X i is the time between counts i 1 and i in a Poisson process, the distribution of X 1 + X 2 + X 3 is Erlang. Exam #3a, Spring 2002 Page 1 of 4 Schmeiser

3 2. (Montgomery and Runger, 6 61) The yield in pounds from a day s production is normally distributed with a mean of 1500 pounds and a variance of pounds squared. Assume that the yields on different days are independent random variables. Let X i be the yield on day i for i = 1, 2,..., 5. (a) Sketch the pdf of X i. Label and scale both axes. Horizontal and vertical axis. Label with x and f X (x ). Sketch a bell curve, centered at µ X = The points of inflection should be one standard deviation from µ X. Here one standard deviation is the square root of the variance, σ X = 100. Scale the vertical axis at zero and somewhere else, probably the mode. The pdf integrates to one, so the value at the mode should be about (The exact value is 1 / ( 2πσ) ) (b) What is the probability that the yield exceeds 1400 pounds on a randomly selected day? (Approximate as necessary.) P(X i > 1400) = P(X i > µ X σ X ) 0.84 using the approximation that about 68% of the normal distribution falls within one standard deviation of the mean. (Half of 68% is 34% to the left of the mean. The right half is 50%, for a total of 84%.) (c) Let p denote your answer to Part (b). What is the probability that production exceeds 1400 pounds all five days of a randomly selected week? (That is, determine P(X 1 > 1400, X 2 > 1400,..., X 5 > 1400).) P(X 1 > 1400, X 2 > 1400,..., X 5 > 1400) = 5 Π i =1 P(X i > 1400) = p 5 Exam #3a, Spring 2002 Page 2 of 4 Schmeiser

4 3. Result: E(XY ) = E(X ) E(Y ). The following proof assumes that X and Y are continuous. Choose exactly one reason for each step. (i) Axiom 1 (ii) Axiom 2 (iii) Axiom 3 (iv) Definition of expected value (v) Density functions integrate to one. (vi) Independence (vii) Total Probability (viii) Definition of pdf (ix) Calculus (No probability result needed) E(XY ) = xy f XY (x, y ) dx dy < (iv) > = xy f X (x ) f Y (y ) dx dy < (vi) > = xfx (x ) dx yfy (y ) dy < (ix) > = E(X )E(Y ) < (iv) > 4. The Weibull cdf is F (x ) = 1 exp[ (x /δ) β ]ifx>0 and zero elsewhere. Write P(2 X 5) in terms of F. The Weibull distribution is continuous, so P(2 X 5) = P(2 <X 5) = P(X 5) P(X 2) = F (5) F (2) Exam #3a, Spring 2002 Page 3 of 4 Schmeiser

5 5. Recall: The exponential cdf is F (x ) = 1 e λx if x>0 and zero elsewhere. The pdf is f (x ) =λe λx if x>0 and zero otherwise. The associated mean value is 1 / λ. (From Montgomery and Runger, Problem 5 114) The CPU of a personal computer has a lifetime that is exponentially distributed with a mean lifetime of six years. (a) What is the value of λ, including its units. λ=1 / 6 (failure per year) (b) Suppose that you have owned this CPU for three years. What is the probability that the CPU fails sometime during the next four years? P(X <7 X>3) = P(X<4) memoryless property = P(X 4) X is continuous = F (4) definition of cdf = 1 e (1 / 6)(4) X is exponential = 1 e 2 / 3 simplify 6. Consider the joint pdf (a, b ) = ab for 0 <a <c and 0 <b <c and zero elsewhere. Determine the value of the constant c. Every pdf must integrate to one, which gives us one equation with the unknown c : Let s rewrite the pdf using the more-common dummy variables x and y : (x, y ) = xy for 0 <x <c and 0 <y <c and zero elsewhere. f XY (x, y ) dx dy = 1 0 c 0 c xy dx dy = 1 c 4 / 4 = 1 c = 2 Exam #3a, Spring 2002 Page 4 of 4 Schmeiser