Lecture 4 Comments 1. Ignore the Continuity Correction in Section 5.6. Homework 1. Let Y be a number between 0 and 1 produced by a computer's random-number-generatoralgorithm. Find the following: (a) P(Y 0.45), Answer: 0.45 (b) P(0.35 < Y < 0.50), Answer: 0.15 (c) P(Y > 0.75), Answer: 0.25 (d) The 75th percentile of the distribution of Y , Answer: 0.75 (e) The value of y such that P(0.12 < Y < y) = 0.13, Answer: 0.25 (f) The value of y such that P(y < Y < 0.75) = 0.30. Answer: 0.45 (5.2.5) 2. Let the random variable X have a distribution given by the rst graph on page 5-9. (a) Check that f(X) is a PDF. (b) Find P(1 < X < 11). Answer: 1 (c) Find f (11). Answer: 0 (d) Find f (1). Answer: 0.20 (5.2.6) 3. Let X be the sum of four randomly generated numbers between 0 and 1, inclusive. It is clear that X is a continuous random variable with range 0 x 4. The graph of the PDF is given near the bottom of page 5-9. (a) Verify that the area of the region between x = 0 and x = 4 is 1. (b) Find P(X > 2). 0.50 (c) True or False: P(X < 1) = 0.25. Comment. False (5.2.7) 4. Let X U (25; 65). Find the following: (a) P(X = 30), Answer: 0 (b) P(30 < X < 40), Answer: 1/4 (c) P(X > 60), Answer: 1/8 (d) P(X 25), Answer: 0 (e) P(X 45) Answer: 1/2. (5.3.1) e 5. The time for a lab assistant, in minutes, to setup is thought to be uniform in the interval [25; 55]. What is the probability that the setup time will exceed 40 minutes? Answer: 0.5 (5.3.4) Remark: The normal distribution is symmetric about its mean. Take the random variable X. Suppose X N( , 2 ). For any number a, P(X > + a) = P(X < - a). In particular, e P(X > ) = P(X < ). Since P(X > ) + P(X < ) = 1, P(X > ) = 0.5. We use the symmetry of the normal distribution in the next problem. 6. Let X N (100; 25). Find the following: (a) P(X < 100) Answer: 1/2, (b) P(X > 100) Answer: 1/2, (c) P(X = 100) Answer: 0, (d) the 99th percentile for this distribution Answer: approx 111.632. (5.4.2) e 7. Reminder: Z is the standard normal random variable. (a) Find P(Z > -1.52). Answer: 0.936 (5.5.2) (b) Find P(-1 < Z < 1). Answer: 0.683 (5.5.3) (c) Find the 90th percentile of the Z distribution. Answer: 1.282 (5.5.4) (d) Find P(Z < -1 or Z > 2). Answer: 0.181 (5.5.5) 1 (e) Find z such that P(Z > z) = 0.80. Answer: -0.842 (5.5.6) (f) Find z such that P(Z < z) = 0.99. Answer: 2.326 (5.5.7) (g) Find z such that P(Z < z) = 0.95. Answer: 1.645 (5.5.9) (h) Find P(Z < -0.94). Answer: 0.174 (i) Find P(1.23 < Z < 2.09). Answer: 0.0910 (j) Find P(Z = 1). Answer: 0 (5.5.10) 8. Let X N (10; 25). Find the following: (a) P(X > 10), Answer: 1/2 (b) P(X > -20), Answer: 1 (c) P(0 < X < 20), Answer: 0.954 (d) P(X = 10), Answer: 0 (e) x so that P(X < x) = 0.10. Answer: 3.592 (5.5.11) e 9. Let X N (50; 16). Find the x value corresponding to the 75th percentile. Answer: 52.698 (5.5.14) e 10. Let X N (20; 4). (a) Fill in the blank: P(X < 18) = P(X > ). Hint: use the symmetry of the normal distribution. Answer: 22 (b) Find the 90th percentile of X. Answer: 22.563 (5.P.8) e 11. In this exercise, you will learn about the 68-95-99.7 rule: Take the normal random variable X N( ; 2 ). (a) What is P( - < X < + )? Hint: convert to Z. Draw the normal curve and shade in the corresponding probability. Answer: 0.683 (b) What is P( - 2 < X < + 2 )? Hint: convert to Z. Draw the normal curve and shade in the corresponding probability. Answer: 0.954 (c) What is P( - 3 < X < + 3 )? Hint: convert to Z. Draw the normal curve and shade in the corresponding probability. Answer: 0.997 e 12. Let X N (15; 1=4). (a) Find P(X > 15). Answer: 0.5 (b) Find the 95th percentile of the distribution. Answer: 15.825 (5.P.11) e 13. Let X N (100; 64). Find the following: (a) and , (b) P(X < ), Answer: 1/2 (c) P(X > ), Answer: 1/2 (d) P(X < 84). Answer: 0.0228 (5.TY.1) e 2
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