Q 1: The number of students using Math lab per day is found in the distribution below. x 6 8 10 12 14 P(x) 0.15 0.3 0.35 0.1 0.1 a) Find the mean for this probability distribution. b) Find the variance for this probability distribution. c) Find the standard deviation for this distribution. d) What is the probability that fewer than 8 or more than 12 use the lab in a given day? Q-2: Suppose that the random variable X has the following cumulative distribution function: x 2 3 4 5 6 7 8 9 10 11 12 F(x) 1/36 3/36 6/36 10/36 15/36 21/36 26/36 30/36 33/36 35/36 1 a) Find the probability distribution of this random variable. b) Find p(5 X 10) c) Find the mean of the random variable X. d) Find the standard deviation of the random variable X. Q 3: A shipment of 7 television set contains 2 defective sets. A hotel makes a random purchase of 3 of the sets. If x is the number of defective sets purchased by the hotel find: a) The probability distribution of x. b) The cumulative distribution function of the random variable x. c) Find p ( 0 x 1) d) Find p( 0 x 2) 1
Q 1: A continuous random variable x that can assume values between x=1 and x=3 has a density function given by f(x)= 1/2 a) Show that the area under the curve is equal to 1. b) Find p(2< x <2.5) c) Find p( x 1.6 ) d) Compute the cumulative distribution function of x, and use it to evaluate p(2 x 2.5 ). Q 2: Suppose x is a continuous random variable with density function given by: f x 1 2 kx 0 x 1 3 0 otherwise 1- Find k. 2- Find p(x 1/ 2 ) and p(1/2<x< 3/4). 3- Find the cumulative distribution function. Q 3: Suppose x is a random variable with density function given by: f x 1 x 2 0 0 x 2 otherwise a) Find the cumulative distribution function. b) Find p(x<1) and p(1<x<1.1) c) Find the mean of the random variable X. d) Find the variance of the random variable X. 2
b n x x nx x; n, p p q, x 0, 1, 2,... Q 1: [3+4+3Marks] From past records, a clothing store has discovered that 30% of people who enter the store will actually make a purchase. Eight people enter the store during a one-hour period. Find the probability that the following number of people will make a purchase. a) Exactly four people b) At least one person c) All eight people Q 2: Suppose x is a binomial random variable with n = 4 and q=2/3. Find p(x=3) a) A fair coin is tossed 8 times. Find the probability of getting at least 2 heads. Q 3 a) Suppose x is a binomial random variable with n = 4 and p=1/3. Find p(x=3) b) A fair coin is tossed 8 times. Find the probability of getting at least 2 heads. g(x; p) = pq x 1, x = 1, 2, 3,.... Q 1 A family decided to continue having children until they have a boy. 1- What is the probability that they will have 4 children? 2- What is the probability that they have at most 4 children. 3
p x; t e t t x! x Q1 During a laboratory experiment, the average number of radioactive particles passing through a counter in 1 millisecond is 4. What is the probability that 6 particles enter the counter in a given millisecond? f x; A, B 1, B A 0, A x B elsewhere Q 1: [3+2+2+3 Marks] Suppose the random variable x is the flight time for a trip from Kuwait to Egypt on a certain airline has been found to vary between 150 minutes and 210 minutes. Moreover, the probability of the actual time falling in any given one-minute interval in this range is the same. In fact it can be assumed that the flight time x for a trip has a uniform distribution on the interval [150, 210]. e) What is the probability density function? f) Find the probability that the flight time for a randomly chosen flight is between 170minutes and 180minutes. g) Find the mean of the random variable X. h) Find the variance of the random variable X. 4
z X Ali s z- score on math exam is -2.53. The average score was 72 and the standard deviation was 5.41. What was his actual score? Q 5: [3+3+4 Marks] A certain brand of automobile tire has a life expectancy that is normally distributed with a mean of = 20000 miles and standard deviation of = 2500miles. a) What is the probability that a randomly chosen tire will last for 20000 miles or more? b) What is the probability that a randomly chosen tire will last for 22500 miles or less? c) What is the probability that a randomly chosen tire will last between 17500 miles and 25000 miles? Hint: use the below table. Z -1.1-1.0 0.0 0.8 1.0 1.35 2.0 p(z<z) 0.1357 0.1587 0.5000 0.7881 0.8413 0.9115 0.9772 Q 5: [4+3+3]Marks] A survey shows that the time spent by shoppers in supermarkets is normally distributed with a mean of 45 minutes and a standard deviation of 12 minutes. Hint: use the below table. Z -1 0.0 0.8 1 1.25 1.35 2 p(z<z) 0.1587 0.5000 0.7881 0.8413 0.8944 0.9115 0.9772 a) What percent of the shoppers at a supermarket will spend between 33 and 57 minutes in the supermarket? 5
b) What is the probability that a randomly chosen shopper will spend between 45 and 69 minutes in the supermarket? c) What is the probability that a random chosen shopper will spend greater than 60 minutes in the supermarket? Q 5: [3+3+4 Marks] The weekly salaries of 5,000 employees of a large corporation are assumed to be normally distributed with mean $450 and standard deviation $40. Hint: use the below table. Z 0.7 0.75 0.8 1 1.29 1.35 2 p(z<z) 0.7580 0.7734 0.7881 0.8413 0.9015 0.9115 0.9772 d) If an employee is selected at random, find the probability that he or she makes less than $480. e) Find the probability that he or she makes between $480 and $530. f) Find the salary below which exist 90% of the employees salaries. 6