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45-733 PROBABILITY AND STATISTICS I Midterm Examination Answers

Probability and Statistics
Name__________________________

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

1. Suppose we have the bivariate discrete probability distributionæ c(3x^{2}y^{2}- xy) x = 1, 2 f(x,y) = ç y = 1, 3 è 0 otherwise

- Find
**c**.

Therefore,**y 1 3 ---------- 1 | 2 24 | 26 x | | 2 |10 102 | 112 | | ---------- 12 126 | 138****c**= 1/138

- Find
**VAR(X) and VAR(Y)**.

**E(X) = å**= 1*(26/138) + 2*(112/138) = 250/138_{i=1,n}å_{j=1,m}x_{i}f(x_{i}, y_{j}) = å_{i=1,n}x_{i}f_{1}(x_{i})

**E(Y) = å**= 1*(12/138) + 3*(126/138) = 390/138_{j=1,m}å_{i=1,n}y_{j}f(x_{i}, y_{j}) = å_{j=1,m}y_{j}f_{2}(y_{j})

**E(X**= 1^{2}) = å_{i=1,n}å_{j=1,m}x_{i}f(x_{i}, y_{j}) = å_{i=1,n}x_{i}f_{1}(x_{i})^{2}*(26/138) + 2^{2}*(112/138) = 474/138

**E(Y**= 1^{2}) = å_{j=1,m}å_{i=1,n}y_{j}f(x_{i}, y_{j}) = å_{j=1,m}y_{j}f_{2}(y_{j})^{2}*(12/138) + 3^{2}*(126/138) = 1146/138

**VAR(X) = E(X**= (474/138) - (250/138)^{2}) - [E(X)]^{2}^{2}= .1529

**VAR(Y) = E(Y**= (1146/138) - (390/138)^{2}) - [E(Y)]^{2}^{2}= .3176

- Are
**X, Y**Independent?

NO.**f(1,1) = 2/138 ¹ f**_{1}(1)*f_{2}(1) = (26/138)*(12/138) » 2.26/138

Probability and Statistics
Name__________________________

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

2. The firm you work for has a small office that employs 10 workers in 2 job categories – one job has a high rate of pay, and the other job has a low rate of pay. There are 4 workers in the highly paid job category and 6 in the lower paid job category. Your boss has received complaints that workers have been discriminated against in the office. In particular, 7 of the 10 workers belong to a particular ethnic group. Five of the 7 workers from this ethnic group are in the lower paid job category and only 2 of the 7 are in the higher paying category. Your boss has asked you to check into this and report back. Just based on the numbers, do you think that there is evidence of discrimination?The probability that we would observe 5 members of the ethnic group in the low paying job and 2 in the higher paying job if job assignment was totally random is:

This probability is high enough for us to conclude that there probably has not been intentional discrimination.æ7öæ3ö ç ÷ç ÷ è5øè1ø 63 P(E) = -------- = ---- = .3 æ10ö 210 ç ÷ è 6ø

Probability and Statistics
Name__________________________

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

3. Suppose we have the bivariate continuous probability distribution:æ c(4x + y) 0 < x < 1 f(x,y) = ç 0 < y < 1 è 0 otherwise

- Find
**c**.

**cò**_{0}^{1}ò_{0}^{1}(4x + y)dxdy = cò_{0}^{1}[(4x^{2}/2 + yx)|_{0}^{1}]dy = cò_{0}^{1}(2 + y)dy = c(2y + y^{2}/2)|_{0}^{1}= c(5/2)

Hence,**c = 2/5**

- Are
**X, Y**Independent?

**f**_{1}(x) = ò_{-¥}^{+¥}f(x,y)dy = ò_{0}^{1}[(8x + 2y)/5]dy = (1/5)[8xy + y^{2}]|_{0}^{1}= (8x + 1)/5

**f**_{2}(y) = ò_{-¥}^{+¥}f(x,y)dx = ò_{0}^{1}[(8x + 2y)/5]dx = (1/5)[4x^{2}+ 2yx]|_{0}^{1}= (4 + 2y)/5

NO. Clearly**f(x,y) ¹ f**_{1}(x)f_{2}(y)

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

4. The probability that 1 percent of the items produced by a certain process are defective is .8, the probability that 5 percent of the items are defective is .1, and the probability that 10 percent of the items are defective is .1. You draw an item randomly from a large lot and find that it is defective. What is the probability that 5 percent of the items are defective?Let

and let

Graphically, this is:

(.1*.05)/(.8*.01 + .1*.05 + .1*.1)= .005/.023 = 5/23

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

5. You are in the heavy equipment rental business. LetSuppose your daily profit,æ .05 x = 0 ç ç .20 x = 5 ç ç .30 x = 10 ç f(x) = ç .25 x = 15 ç ç .15 x = 20 ç ç .05 x = 25 ç è 0 otherwise

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

6. Three construction contracts are to be randomly assigned to 3 firms – your firm and your two competitors. If each contract will yield you a profit of $100,000, what is your expected profit?This is a "boxes and balls" problem.

Let

Hence the expected profit is $100,000.

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

7. You draw 4 cards randomly from a deck of 52 standard playing cards. What is the probability that exactly one suit is missing (e.g., §§ ¨ ª )?æ4öæ3öæ13öæ13öæ13ö ç ÷ç ÷ç ÷ç ÷ç ÷ è3øè1øè 2øè 1øè 1ø P(E) = ---------------- æ52ö ç ÷ è 4ø

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

8. Suppose we have the discrete probability functionFindæ .25 x = 1 ç ç .05 x = 2 ç f(x) = ç .20 x = 3 ç ç .50 x = 4 ç è 0 otherwise

æ 0 x < 1 ç ç .25 1 £ x < 2 ç F(x) = ç .30 2 £ x < 3 ç ç .50 3 £ x < 4 ç è 1 x ³ 4

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

9. Suppose we have the following circuit with 6 relays. The probability that any relay will work when it is activated is .8 and the failures of the relays are independent. What is the probability that current will flow from A to B when the relays are activated?|--1-----2-------| | | A--|-----3----------|---B | ---5--- | |--4----| |--| ---6---

For current to flow from A to B at least one path has to be complete. Hence

Spring 1999 Flex-Mode and Flex-Time 45-733

Midterm

Keith Poole

(10 Points)

10. Suppose we have the bivariate discrete probability distributionæ (2x^{2}+ y^{2}+ 1)/54 x = 0, 1, 2 f(x,y) = ç y = 0, 1, 2 è 0 otherwise

- Find
**P(X > 0 Ç Y > 0)**.

**y 0 1 2 ------------- 0 | 1 2 5 | 8 | | x 1 | 3 4 7 | 14 | | 2 | 9 10 13 | 32 | | ---------------- 13 16 25 | 54****P(X > 0 Ç Y > 0)**= (4 + 7 + 10 + 13)/54 = 34/54

- Find
**P(X ³ 1 | Y ³ 1)**

**P(X ³ 1 | Y ³ 1) = P(X ³ 1 Ç Y ³ 1)/P(Y ³ 1)**=

(4 + 7 + 10 + 13)/(2 + 5 + 4 + 7 + 10 + 13) = 34/41