CE 21- Problem Set 1

WRITE ANSWERS IN A BLUE BOOK. SUBMIT BEFORE THE EXAM. NO PROBLEM SET, NO EXAM.

1. How many integers between 100 and 999 have distinct digits? How many of those are odd numbers?

2. To keep the monkey entertained, Tarzan gives Cheetah the following letters from a scrablle set to play with:

AAA EE I J K L NN R T Z

What is the probability that Cheetah (who can’t spell) rearranges the letters at random and forms the following

sequence:

TARZAN LIKE JANE

3. In a recent survey, x percent of teens said that they were excited to watch the latest Harry Potter movie while the

rest were excited to watch “New Moon”. Among those who preferred Harry Potter, 37% were male and 63% were

female. For those excited to watch “New Moon”, 28% were male and 72% were female.

a. If the probability that a teenager prefers “New Moon” given that she is a female is 72/149, find x.

b. What is the probability that a randomly selected teenager is male?

c. What is the probability that a randomly selected teenager prefers Harry Potter and is female?

4. A stock solution of hydrochloric acid (HCl) supplied by a certain vendor contains small amounts of several

impurities, including copper and nickel. Let X denote the amount of copper and let Y denote the amount of nickel,

in parts per ten million, in a randomly selected bottle of solution. Assume that the joint probability density function

of X and Y is given by

( ) { ( )

a. Find the value of the constant c so that f(x,y) is a joint density function.

b. Compute the marginal density function fx(x).

c. Compute the conditional density function fY|X (y|x).

d. Compute the conditional expectation E(Y|X=0.4).

e. Are X and Y independent? Explain.

5. An engineer bought a box containing ten AAA batteries from a sidewalk vendor. These batteries did not undergo

quality control testing, and were thus sold at a very cheap price. Four batteries were 90% effective, three batteries

were 75% effective, and the rest were not effective at all.

The engineer gets two batteries at random from the box. Let X = the 90% effective batteries and Y = the 75%

effective batteries.

a. Complete the joint probability mass function of X and Y. Show all solutions.

y

X 0 1 2

0 0.0667

1 0.2667

2 0

b. Solve for the covariance of X and Y. From the value you will obtain, what conclusion can you make regarding

the relationship between X and Y?