Chapter 2: Random Variables (Cont d)

Transcription

1 Civil Engineering Department: Engineering Statistics (ECIV 005) Chapter : Random Variables (Cont d) Section.6: Combinations and Functions of Random Variables Problem (): Suppose that the random variables X, Y, and Z are independent with E(X) =, Var(X) = 4, E(Y ) = 3, Var(Y ) =, E(Z) = 8, and Var(Z) = 7. Calculate the expectation and variance of the following random variables. (a) 3X + 7 (b) 5X 9 (c) X + 6Y (d) 4X 3Y (e) 5X 9Z + 8 (f) 3Y Z 5 (g) X + Y + 3Z (h) 6X + Y Z + 6 (problem.6. in textbook) Engr. Yasser M. Almadhoun Page

2 Civil Engineering Department: Engineering Statistics (ECIV 005) Problem (): A machine part is assembled by fastening two components of type A and one component of type B end to end. Suppose that the lengths of components of type A have an expectation of 37.0 mm and a standard deviation of 0.7 mm, whereas the lengths of components of type B have an expectation of 4.0 mm and a standard deviation of 0.3 mm. What are the expectation and variance of the length of the machine part? (problem.6.4 in textbook) Engr. Yasser M. Almadhoun Page

3 Civil Engineering Department: Engineering Statistics (ECIV 005) Problem (3): The weight of a certain type of brick has an expectation of. kg with a standard deviation of 0.03 kg. (a) What are the expectation and variance of the average weight of 5 bricks randomly selected? (b) How many bricks need to be selected so that their average weight has a standard deviation of no more than kg? (problem.6.6 in textbook) X X X X X X Engr. Yasser M. Almadhoun Page 3

4 Civil Engineering Department: Engineering Statistics (ECIV 005) Problem (4): Suppose that the random variable X has a probability density function f (x) = x for 0 x. Find the probability density function and the expectation of the random variable Y in the following cases. (a) Y = X 3 (c) Y = /( + X) (problem.6.8 in textbook) (b) Y = X (d) Y = X E(y) = yf Y (y) dy = y 3 y /3 dy = Y = X E(y) = yf(y) dy = y4y 3 0 dy = 0.8 +X y y y y Engr. Yasser M. Almadhoun Page 4

5 Civil Engineering Department: Engineering Statistics (ECIV 005) y + y 3 E(y) = yf(y) dy = y ( y + y 3) dy = Y = X x ln(y) ln() ln(y) ln() ( ln(y) ln() ) ( ln(y) ln() ) E(y) = yf(y) dy = y (( ln(y) ln() ) ) dy =.6 Problem (5): The random variable X has an expectation of 77 and a standard deviation of 9. Find the values of a and b such that the random variable Y = a + bx has an expectation of 000 and a standard deviation of 0. (problem.6.4 in textbook) Engr. Yasser M. Almadhoun Page 5

6 Civil Engineering Department: Engineering Statistics (ECIV 005) Problem (6): Suppose that components are manufactured such that their heights are independent of each other with μ = and = 0.3. (a) What are the mean and the standard deviation of the average height of five components? (b) If eight components are stacked on top of each other, what are the mean and the standard deviation of the total height? (problem.6.5 in textbook) Problem (7): Suppose that the impurity levels of water samples taken from a particular source are independent with a mean value of 3.87 and a standard deviation of 0.8. (a) What are the mean and the standard deviation of the sum of the impurity levels from two water samples? (b) What are the mean and the standard deviation of the sum of the impurity levels from three water samples? (c) What are the mean and the standard deviation of the average of the impurity levels from four water samples? Engr. Yasser M. Almadhoun Page 6

7 Civil Engineering Department: Engineering Statistics (ECIV 005) (d) If the impurity levels of two water samples are averaged, and the result is subtracted from the impurity level of a third sample, what are the mean and the standard deviation of the resulting value? (problem.6. in textbook) X +X +X 3 +X 4 4 X +X +X 3 +X 4 4 E(X )+E(X )+E(X 3 )+E(X 4 ) Var(X )+Var(X )+Var(X 3 )+Var(X 4 ) 6 (0.8) +(0.8) +(0.8) +(0.8) 6 X +X +X 3 +X X 3 X +X X 3 X +X X 3 X +X X 3 E(X )+E(X ) X 3 Var(X )+Var(X ) 4 (0.8) +(0.8) Engr. Yasser M. Almadhoun Page 7

8 Civil Engineering Department: Engineering Statistics (ECIV 005) Problem (8): An investment in company A has an expected return of $30,000 with a standard deviation of $4000. An investment in company B has an expected return of $45,000 with a standard deviation of $3000. If investments are made in both companies, what are the expectation and standard deviation of the total return? (problem.6.4 in textbook) 5,000,000 Problem (9): A continuous random variable X that can assume values between x = and x = 3 has a density distribution function given by f(x) = 0.5x: (a) Find P( X.5). (b) Find the interquartile range. (c) Calculate E(X). (d) Calcularte the standard deviation of the random variable X. (e) Find the value of x that is exceeded by only 5% of the data. (f) If Y = X + : i. Find the value of E(Y) and Var(Y). ii. Find the pdf of Y. (Question 4: in Midterm Exam 0/06).5.5 P( X.5) = f(x) dx = 0.5x dx = 0.85 Engr. Yasser M. Almadhoun Page 8

9 Civil Engineering Department: Engineering Statistics (ECIV 005) x 0.5y dy = (x ) = 0.75 x 0.5y dy = (x ) = E(X) = xf(x) dx = x(0.5x)dx =.667 Var(X) = = (x E(X)) f(x) dx = E(X ) (E(X)) = x (0.5x) dx (E(X)) 3 = x (0.5x) dx (.667) = 5.0 (.667) = Engr. Yasser M. Almadhoun Page 9

10 Civil Engineering Department: Engineering Statistics (ECIV 005) Var(X) = = = x 0.5y dy = (x ) = 0.75 X = Y x F X (x) = 0.5y dy = 0.5 (x ) F X (x) = 0.5x 0.5 y b F Y (y) = F X ( ) = 0.5 (Y a ) 0.5 F Y (y) = 0.5 ( Y ) 0.5 F Y (y) = (Y Y + ) 0.5 Engr. Yasser M. Almadhoun Page 0

11 Civil Engineering Department: Engineering Statistics (ECIV 005) F Y (y) = Y 0.5 Y 3 3 f Y (y) = df(y) dx = Y f Y (y) = Y for 3 Y 7 Problem (0): Assume that Y = -5X 7, and given that E(X) = -4, and E(X ) = 30, calculate: (a) Var(X). (b) Var(Y). (c) E(Y 3). (Question : in Midterm Exam 004) Engr. Yasser M. Almadhoun Page

12 Civil Engineering Department: Engineering Statistics (ECIV 005) Problem (): Suppose tha probability density function of the random variable X is: f(x) = 3e 3x x 0 and if Y = -X + : 0.0 x < 0 (a) Find the value of E(Y). (b) Find the value of Var(Y). (Question 4: in Midterm Exam 004) E(X) = xf(x) dx = 0 3xe 3x dx E(Y) = E(X) + = x(3e 3x )dx + 0 Var(X) = = (x E(X)) f(x)dx = E(X ) (E(X)) = x (3e 3x )dx ( x(3e 3x )dx) = x (3e 3x )dx 0 ( x(3e 3x ) dx) 0 Var(Y) = ( ) Var(X) Engr. Yasser M. Almadhoun Page

13 Civil Engineering Department: Engineering Statistics (ECIV 005) Problem (): The probability mass function of the number of calls passing a given switchboard within minute is given as follows: (a) What is the expected number of calls passing the switchboard wihin a minute? (b) What is the variance of the number of calls passing the switchboard wihin a minute? (c) If the number of calls passing the switchboad in any two different minutes are independent, what are the expectation and the variance of the number of calls passing the switchboard wihin hour? (Question : in Final Exam?) E(X) = x i p i Engr. Yasser M. Almadhoun Page 3

14 Civil Engineering Department: Engineering Statistics (ECIV 005) Engr. Yasser M. Almadhoun Page 4