Chapter 2: Random Variables (Cont d)

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1 Chapter : Random Variables (Cont d) Section.4: The Variance of a Random Variable Problem (1): Suppose that the random variable X takes the values, 1, 4, and 6 with probability values 1/, 1/6, 1/, and 1/6, respectively. (a) Find the expectation of X. (b) Find the variance of X using the formula: Var(X) = E((X E(X)) ). (c) Find the variance of X using the formula: Var(X) = E(X ) (E(X)). (problem.4.1 in textbook) E(X) = x i p i σ σ Engr. Yasser M. Almadhoun Page 1

2 Problem (): An office has four copying machines, and the random variable X measures how many of them are in use at a particular moment in time. Suppose that: P(X = 0) = 0.08, P(X = 1) = 0.11, P(X = ) = 0.7, P(X = ) = 0. and P(X = 4) = 0.1. Calculate the variance and standard deviation of the number of copying machines in use at a particular moment. (problem.4. in textbook) σ σ Var(X) = σ = Problem (): A company has five warehouses, only two of which have a particular product in stock. A salesperson calls the five warehouses in a random order until a warehouse with the product is reached. Let the random variable X be the number of calls made by the salesperson. Calculate the variance and standard deviation of the number of warehouses called by the salesperson. (problem.4. in textbook) x i p i = = = 1 10 Engr. Yasser M. Almadhoun Page

3 σ σ Var(X) = σ 1.0 = 1.0 Problem (4): Consider a random variable with a probability dinsity function of: f(x) = f(x) = 0 1 x ln (1.5) for 4 x 6 elsewhere (a) What is the variance of this random variable? (b) What is the standard deviation of this random variable? (c) Find the upper and lower quartiles of this random variable. (d) What is the interquartile range? (problem.4.5 in textbook) Var(X) = σ = (x E(X)) f(x) dx = E(X ) (E(X)) = x 1 ( ) dx (E(X)) x ln (1.5) 6 = x 1 ( x ln (1.5) ) dx (4.94) 4 Engr. Yasser M. Almadhoun Page

4 = (4.66) (4.94) = 0.6 σ Var(X) = σ = 0.6 = x 1 dy = 0.75 y ln(1.5) 4 x 1 dy = 0.5 y ln(1.5) Problem (5): Consider a random variable with a cumulative distribution function of: F(x) = x 16 for 0 x 4 (a) What is the variance of this random variable? Engr. Yasser M. Almadhoun Page 4

5 (b) What is the standard deviation of this random variable? (c) Find the upper and lower quartiles of this random variable. (d) What is the interquartile range? (problem.4.6 in textbook) f(x) = x 8 for 0 x 4 Var(X) = σ = (x E(X)) f(x) dx = E(X ) (E(X)) = x ( x ) dx (E(X)) 8 4 = x ( x 8 ) dx (.67) 0 = (8.0) ( 8 ) = 8 9 σ Var(X) = σ = 8/9 = x 16 = 0.75 Engr. Yasser M. Almadhoun Page 5

6 x 16 = 0.5 Problem (6): The time taken to serve a customer at a fast-food restaurant has a mean of 75.0 seconds and a standard deviation of 7. seconds. Use Chebyshev s inequality to calculate time intervals that have 75% and 89% probabilities of containing a particular service time. (problem.4.10 in textbook) Engr. Yasser M. Almadhoun Page 6

7 Problem (7): A machine produces iron bars whose lengths have a mean of cm and a standard deviation of 0.5 cm. Use Chebyshev s inequality to obtain a lower bound on the probability that an iron bar chosen at random has a length between cm and cm. (problem.4.11 in textbook) 1 1 (.5) 1 1 (.5) Problem (8): A continuous random variable has a probability density function: f (x) = Ax.5 for x. (a) What is the value of A? (b) What is the expectation of the random variable? (c) What is the standard deviation of the random variable? (d) What is the median of the random variable? (problem.4.14 in textbook) Engr. Yasser M. Almadhoun Page 7

8 Area under the pdf = f (x) dx = 1.0 Ax.5 = 1.0 [ Ax.5.5 ] = 1.0 ( A().5.5 ) (A().5.5 ) = 1.0 f (x) = x.5 for x. E(X) = xf(x) dx = x( x.5 ) dx =.58 Var(X) = σ = (x E(X)) f(x) dx = E(X ) (E(X)) = x ( x.5 ) dx (E(X)) = x ( x.5 ) dx (.58) = (.58) = σ Var(X) = σ = = Engr. Yasser M. Almadhoun Page 8

9 x.5 dx = 0.5 [ x.5 ] = 0.5 Problem (9): In a game, a player either loses $1 with a probability 0.5, wins $1 with a probability 0.4, or wins $4 with a probability 0.5. What are the expectation and the standard deviation of the winnings? (problem.4.15 in textbook) σ σ Var(X) = σ.8475 = Problem (10): A random variable X has a distribution given by the probability density function f (x) = (1 x)/ with a state space 1 x 1. (a) What is the expected value of X? (b) What is the standard deviation of X? (c) What is the upper quartile of X? (problem.4.18 in textbook) E(X) = xf(x) dx = 1 1 x 1 x dx = 1 Engr. Yasser M. Almadhoun Page 9

10 Var(X) = σ = (x E(X)) f(x) dx = E(X ) (E(X)) = x ( 1 x ) dx (E(X)) 1 = x ( 1 x ) dx ( 1 ) 1 = 1 ( 1 ) = 9 σ Var(X) = σ = /9 = x 1 1 y dy = 0.75 Problem (11): Three independent events A, B and C have probabilities 1/, /, /4, respectively. Let X be the number of these events that occure, where: 0 X. Find the following: (a) P(X = 0) (b) P(X = 1) (c) P(X = ) (d) P(X = ) (e) E(X) (f) Var(X) Engr. Yasser M. Almadhoun Page 10

11 (Question 5: in Midterm Exam 005) σ Engr. Yasser M. Almadhoun Page 11

12 Problem (1): Suppose the comulative distribution function of the ransdom variable X is: 0.0 x < F(X) = Ax x < B x (a) Find the value of A and the expected value of B? (b) What is the probability P( 1 X 1)? (c) Calculate the variance of this random variable? (d) What is the interquartile range? (e) Sketch the probability density function and the cumulative distribution function? (Question : (15 points) in Midterm Exam 007) Area under the pdf = f(x) dx = A 1.0 = [Ax] 1.0 = A( ( )) dx = x < F(X) = 0.5x x < 1.0 x Engr. Yasser M. Almadhoun Page 1

13 E(X) = x f(x) dx = x(0.5) dx = 0.00 Var(X) = σ = (x E(X)) f(x) dx = E(X ) (E(X)) = x (0.5x + 0.5) dx (E(X)) 6 = x (0.5x + 0.5) dx (0.00) 4 = 4/ (0.00) = 4/ 0.0 x < F(X) = 0.5 x < 0.0 x Engr. Yasser M. Almadhoun Page 1

14 X F(x) 0 x < F(X) = 0.5x x < 1.0 x X F(x) Engr. Yasser M. Almadhoun Page 14

15 Problem (1): The pdf of a random variable X is shown: (a) Calculate the median of X. (b) Calculate E(X). (c) Graph the cdf on the chart below. (d) On the cdf graph in part c, locate the upper quartile and the lower quartile and find their values from the cdf graph. (Question : (6 points) in Midterm Exam 010) b x =?! f(x) = x 6 for x 5 Engr. Yasser M. Almadhoun Page 15

16 f(x) = 1 for 5 x 7 f(x) = x 6 1 x 5 5 x 7 E(X) = xf(x) dx = x ( x 6 ) dx + x 1 = dx = x F(x) = y dy = [ 6 y y ] dy = [ y x ] = x 4 5 x x = x 1 x + 4 x 5 5 x 7 Engr. Yasser M. Almadhoun Page 16

17 Problem (14): A random variable X takes values between 0 and with a cumulative distribution function: F(x) = A + Be x (a) Find the value of both A and B. (b) What is the probability that the random variable X takes a value between 1 and 7. (c) Find the probability density function. (d) Calculate the median of this random variable. (e) Compute the interquartile range. (f) Compute the variance. (Question 4: (7 points) in Midterm Exam 010) F( ) = 1.0 A + B e x = 1.0 A = 1.0 F(0) = 0.0 A + Be x = 0.0 Engr. Yasser M. Almadhoun Page 17

18 1 + B = 0.0 B = 1.0 F(x) = 1 e x for 0 x = = P(1 X 7) = F(X = 7) F(X = 1) = (1 e 7 ) (1 e 1 ) = = f(x) = df(x) dx = d(1 e x ) = e x for 0 x dx f(x) = e x for 0 x 1 e x = 0.50 Engr. Yasser M. Almadhoun Page 18

19 1 e x = e x = 0.5 Var(X) = σ = (x E(X)) f(x)dx = E(X ) (E(X)) = x (e x )dx 0 ( x(e x )dx) 0 Engr. Yasser M. Almadhoun Page 19

Version A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.

Version A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise. Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x