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1. (Level 2) Consider randomly picking a married couple from Minnesota. Let W denoted the height of the wife (in inches) and let

 


 

H denoted the height of the husband (in inches). Pearson and Lee claimed the population scatter plot is approximately

 

?football shaped? and published the following approximations for the expected values, standard deviations and correlation

 

between H and W:

 

EW = 63,

 

sd(W) = 2.5

 

(10)

 

EH = 68,

 

sd(H) = 2.7

 

(11)

 

? = 0.25

 

(12)

 


 

(a) If W = 66 what do you predict the husbands height to be?

 

(b) Approximate the percentage of married women from Minnesota are over 5 feet 8 inches tall?

 

(c) Of the women who are married to men 6 feet tall, what percentage are over 5 feet 8 inches tall (give an approximation)?

 

2.

 


 

(Level 2) Consider two random variables X,Y whose correlation is ? = 0.7 (and the joint PMF is bivariate Gaussian). Predict the

 

z-score for Y in each of the following cases

 


 

(a) You observe a z-score for X of .2.

 

(b) You observe a z-score for X of ?.7

 

(c) You observe that (X ? EX)/sd(X) = 1.2

 

(d) You observe that (X ? EX)/sd(X) = 3.2

 

(e) You observe that (X ? EX)/sd(X) = 0

 

(f) You observe that X = 4 (assuming EX = 2 and sd(X) = 5.5).

 

(g) You observe that X = ?2 (assuming EX = 0 and var(X) = 4).

 

(h) You observe that X is at the 90th percentile (assuming X ? N(0,1))

 

(i) You observe that X is at the 30th percentile (assuming X ? N(0,1))

 

(j) You observe that X is at the 90th percentile (assuming X ? N(?19,10))

 

(k) You observe that X is at the 30th percentile (assuming X ? N(4,4))

 

3.

 


 

(Level 2) Consider two random variables X,Y whose correlation is ? = ?.1 (and the joint distribution of bivariate normal).

 

Predict the z-score for X in each of the following cases

 


 

(a) You observe a z-score for Y of .2.

 

(b) You observe a z-score for Y of ?.7

 

(c) You observe that (Y ? EY )/sd(Y ) = 1.2

 

(d) You observe that (Y ? EY )/sd(Y ) = 3.2

 

(e) You observe that (Y ? EY )/sd(Y ) = 0

 

(f) You observe that Y = 4 (assuming EY = 2 and sd(Y ) = 5.5).

 


 

1

 


 

(g) You observe that Y = ?2 (assuming EY = 0 and var(Y ) = 4).

 

(h) You observe that Y is at the 90th percentile (assuming Y ? N(0,1))

 

(i) You observe that Y is at the 30th percentile (assuming Y ? N(0,1))

 

(j) You observe that Y is at the 90th percentile (assuming Y ? N(?19,10))

 

(k) You observe that Y is at the 30th percentile (assuming Y ? N(4,4))

 

4.

 


 

(Level 2) Consider two bivariate random variables X,Y . Suppose EX = 10, EY = 5, sd(X) = 15, sd(Y ) = 2 and

 

? = 0.9. Predict for Y in each of the following cases

 


 

(a) You observe a z-score for X of .2.

 

(b) You observe a z-score for X of ?.7

 

(c) You observe that (X ? EX)/sd(X) = 1.2

 

(d) You observe that (X ? EX)/sd(X) = 3.2

 

(e) You observe that (X ? EX)/sd(X) = 0

 

(f) You observe that X = 4.

 

(g) You observe that X = ?2.

 

(h) You observe that X is at the 90th percentile. (i) You observe that X is at the 30th percentile.

 

5.

 


 

(Level 2) Consider two bivariate normal random variables X,Y . Suppose EX = 10, EY = 5, sd(X) = 15, sd(Y ) = 1 and ? = 0.

 

Predict for Y when observing X = ?5. What would you expect your prediction error to be?

 


 

6.

 


 

(Level 2) Let W and Q be two bivariate normal random variables such that

 

EW = 5, sd(W) = 1 (13) EQ = 10, sd(Q) = 2 (14) ? = .2

 


 

(15)

 


 

What is the approximate expected value of W when Q = 15

 

(ie what is

 

7.

 


 

= 15))? What is the approximate probability that W > 5 given Q = 15.

 


 

Let W and Q be two bivariate normal random variables such that

 

EW = 5, sd(W) = 1, (16) EQ = ?10, sd(Q) = 2, (17) ? = .8 (18)

 

What is the approximate probability that W > 5 given Q = ?7.

 


 

8.

 


 

Let W and Q be two bivariate normal random variables such that

 

EW = 5,

 

sd(W) = 1,

 

EQ = ?10,

 

sd(Q) = 2,

 

? = .2

 

What is the approximate probability that W > 5 given Q = ?7.

 


 

9.

 


 

(Level 2) Let W and Q be two bivariate normal random variables such that

 

EW = ?1,

 

sd(W) = 10

 

(19)

 

EQ = ?10,

 

sd(Q) = 2

 

(20)

 

? = ?.9.

 

(21)

 


 

2

 


 

Predict the value of Q when you observe W = ?10. What is the typical error for your prediction? Approximate the probability

 

that your prediction falls within 2 of the true value of Q.

 

10. (Level 2) Let X and Y be two random variables such that

 


 

EX = ?1, sd(X) = 10 (22) EY = ?10, sd(Y ) = 2 (23) ? = ?.9

 


 

(24)

 


 

Assume the joint distribution of (X,Y ) is bivariate normal. Fill in the blanks:

 


 

(a) The conditional PMF, p(y|X = 0), is approximately normal with meanand variance .

 

(b) The conditional PMF, p(x|Y = ?8), is approximately normal with meanand variance

 

.

 


 

(c) Given that X = 5 we predict that Y will be and our typical prediction will be off by about

 


 

or so.

 


 

11. Suppose X and Y are two bivariate normal random variables with correlation ? = ?0.5. If you observe that X has a z-score of

 


 

1.5 what is your predicted z-score for Y ?

 

12. (Level 2) Elizabeth runs a popular television show on NBC. She notices that the on-camera time of her two stars, Tracy and

 


 

Jenna, are both approximately normally distributed

 


 

3

 


 

13.

 

14.

 


 

(a)

 

15.

 


 

(a)

 

(b)

 


 

 


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