# Exam 2_SOLUTION

Problems1. (24 points total, 4 points each) True/False Questions. Circle either True or False.

(a) True or False: A probability density function fX(x) for a continuous random variable X can take values greater than 1.

(b) True or False: A cumulative distribution function (CDF) FZ(z) can take values greater than 1.

(c) True or False: The integral of the probability density function fX(x), over −∞ < x < ∞, is always equal to 1.

(d) True or False: The probability that X = 1, for a discrete random variable X, is the probability mass function fX(x), evaluated at x = 1.

(e) True or False: EX2is equal to (E [X])2 for any continuous random variable X.

(f) True or False: For a Binomial random variable K, which represents the number of successes in n trials where each trial is a success with probability p, the expected value E [K] is equal to np.

2. (6 points) Circle the one correct answer: Let X be a random variable with standard deviation σX and mean µX. What is NOT a valid expression for the variance of X? (a) EX2 (b) (σX)2 (c) Eh(X − µX)2i (d) EX2− (µX)2

3. (6 points) Circle the one correct answer: We ﬂip a fair coin 10 times. What is the probability of getting 5 tails? (a) 10 5 1 210 (b) 10! 5! 1 210 (c) 0.5 (d) 0.1 (e) 1 210

4. (6 points) Circle the one correct answer: For continuous random variables X and Y , we ﬁnd that E [X] = 1.0, E [Y ] = 4.0, E [XY ] = 5.0, EX2= 1.5, and EY 2= 20.0. What is the correlation coeﬃcient ρXY ?

(a) 0.354 (b) 0.5 (c) 0.707 (d) 1.0 (e) 2.0

5. (16 points total) Random variable X is Gaussian with mean 2.0 and standard deviation 4.0.

(a) (7 points) Find P [X 7.5].

Answer:

(b) (9 points) Find a value d such that P [2.0 − d < X < 2 + d] = 0.90.

Answer:

ECE 3530 / CS 3130 Spring 2014 5

6. (14 points) Let X have the pdf:

f(x) = 3 x4 , x 1 0, o.w.

(a) Find Var [X].

Answer:

7. (28 points total) There are three true/false questions, and one multiple choice question on an exam. Let X be the number of true/false questions a student answers correctly, and let Y be the number of multiple choice questions the student answers correctly. From data, we determine the joint pmf fX,Y (x,y) is as follows:

fX,Y (x,y) X = 0 X = 1 X = 2 X = 3 Y = 0 0.04 0.08 0.06 0.07 Y = 1 0.02 0.12 0.30 0.31

(a) (6 points) What is fX(3)?

Answer:

(b) (6 points) What is fX|Y (x = 3|y = 1)?

Answer:

(c) (12 points) What is the covariance of X and Y ?

Answer:

(d) (4 points) True or False: X and Y are independent. Answer:

(a) True or False: A probability density function fX(x) for a continuous random variable X can take values greater than 1.

(b) True or False: A cumulative distribution function (CDF) FZ(z) can take values greater than 1.

(c) True or False: The integral of the probability density function fX(x), over −∞ < x < ∞, is always equal to 1.

(d) True or False: The probability that X = 1, for a discrete random variable X, is the probability mass function fX(x), evaluated at x = 1.

(e) True or False: EX2is equal to (E [X])2 for any continuous random variable X.

(f) True or False: For a Binomial random variable K, which represents the number of successes in n trials where each trial is a success with probability p, the expected value E [K] is equal to np.

2. (6 points) Circle the one correct answer: Let X be a random variable with standard deviation σX and mean µX. What is NOT a valid expression for the variance of X? (a) EX2 (b) (σX)2 (c) Eh(X − µX)2i (d) EX2− (µX)2

3. (6 points) Circle the one correct answer: We ﬂip a fair coin 10 times. What is the probability of getting 5 tails? (a) 10 5 1 210 (b) 10! 5! 1 210 (c) 0.5 (d) 0.1 (e) 1 210

4. (6 points) Circle the one correct answer: For continuous random variables X and Y , we ﬁnd that E [X] = 1.0, E [Y ] = 4.0, E [XY ] = 5.0, EX2= 1.5, and EY 2= 20.0. What is the correlation coeﬃcient ρXY ?

(a) 0.354 (b) 0.5 (c) 0.707 (d) 1.0 (e) 2.0

5. (16 points total) Random variable X is Gaussian with mean 2.0 and standard deviation 4.0.

(a) (7 points) Find P [X 7.5].

Answer:

(b) (9 points) Find a value d such that P [2.0 − d < X < 2 + d] = 0.90.

Answer:

ECE 3530 / CS 3130 Spring 2014 5

6. (14 points) Let X have the pdf:

f(x) = 3 x4 , x 1 0, o.w.

(a) Find Var [X].

Answer:

7. (28 points total) There are three true/false questions, and one multiple choice question on an exam. Let X be the number of true/false questions a student answers correctly, and let Y be the number of multiple choice questions the student answers correctly. From data, we determine the joint pmf fX,Y (x,y) is as follows:

fX,Y (x,y) X = 0 X = 1 X = 2 X = 3 Y = 0 0.04 0.08 0.06 0.07 Y = 1 0.02 0.12 0.30 0.31

(a) (6 points) What is fX(3)?

Answer:

(b) (6 points) What is fX|Y (x = 3|y = 1)?

Answer:

(c) (12 points) What is the covariance of X and Y ?

Answer:

(d) (4 points) True or False: X and Y are independent. Answer:

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