Midterm 1(4)

1. 3 % of details are defective. Four details are at random taken. Find the probability, that among them: а) two defective; б) one defective; в) at least one defective. (3)

2. What it is more probable to win against the equivalent opponents: 2 parties from 3 or 4 parties from 7? (3)

3. 10 independent shoots on targets, probability of hit in which 0,9, are made. Find the most probable number a) of hits b) of miss. (3)

4. A plant has produced 4000 details. The probability that detail is defective is to 0,0001. Find the probability of а) 3 details are defective; б) not more then 5 details are defective ; в) at least one detail is defective. (3)

5 A production manager knows that 5% of components produced by a particular manufacturing process have some defect. Six of these components, whose characteristics can be assumed to be independent of each other, were examined. What is the probability that none of these components has a defect?

A) 0.05⁶; B) 0.95⁶; C) 0.95³0.05³; D) 0.95³0.05³; E) 0.95⁴0.05². {Correct answer} = B

6 .Compute the expected value of the discrete random variable described by throwing a fair dice.

A) 7/12; B) 7/3; C) 21/8; D) 5/3; E) 7/2. {Correct answer} = E

7 Suppose that the random variable T has the following probability distribution:

t	0	1	2
P(T = t)	0.5	0.3	0.2

Compute E(T), the mean of the random variable T.

A) 1.00; B) 1.20; C) 0.70; D) 2.1; E) 1.1. {Correct answer} = C

8 Let X be continuous random variable with probability density function

Find the expected value of random variable X.

A) 12/7; B) 13/3; C) 27/4; D) 28/9; E) 19/3. {Correct answer} = D

9. - Let be continuous random variable with probability distribution function

Find the expected value -of random variable

À. ; В. ; Ñ. ;

Ä. ; Å. .

10. Find the vairiance of random variable

	-2	1	2
Ð	0,1	0,6	0,3

À. 0,5; Â. 1,67; Ñ. 4,71;

Ä. 1,2; Å. 4,7.

11. Let be continuous random variable with probability distribution function

Find the expected value and the variance of random variable

À. ; Â. ;

Ñ.; Ä. ;

Å. .

12. , . Find .

13..Let be continuous random variable with probability density function . Then

À. ; Â. ; Ñ. ;

Ä. ; Å. .

14. Let X be continuous random variable with probability density function

Find

À. 3/4; Â. 1/2; Ñ. 1/6; Ä. 1 Å. 0.

;15. Let be continuous random variable with probability distribution function

Find probability density function

À. в.

с д.

e. Д½рыс жауабы к¼рсетiлмеген

16.Let be continuous random variable with probability distribution function

.

À. ; Â. ; Ñ. ;

Ä. ; Å. ;

Let X be continuous random variable with probability density function

-,?

À. 3/4; Â. 4; Ñ. 1; Ä. 3; Å. 2.

2. [Continuous-type random variables II]

The pdf of a random variable X is given by:

if E[X]=5/8, find a and b.

3. [Uniform and exponential distribution I]

(a) In a new subdivision, along a street of some finite length L, a fire station will be built. If fires will occur at points uniformly chosen on (0, L), where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to minimize E[|X − a|], when X is uniformly distributed over (0, L)

(b) Now suppose the street is of infinite length- stretching from point 0 outward to ∞. If the distance of a fire from point 0 is exponentially distributed with rate λ, where should the fire station now be located?

4. [Uniform and exponential distribution II]

A factory produced two equal size batches of radios. All the radios look alike, but the lifetime of a radio in the first batch is uniformly distributed from zero to two years, while the lifetime of a radio in the second batch is exponentially distributed with parameter λ = 0.1(years)−1.

(a) Suppose Alicia bought a radio and after five years it is still working. What is the conditional

probability it will still work for at least three more years?

(b) Suppose Venkatesh bought a radio and after one year it is still working. What is the conditional probability it will work for at least three more years?

5. [Poisson process]

A certain application in a cloud computing system is accessed on average by 15 customers per minute.

Find the probability that in a one minute period, three customers access the application in the first ten seconds and two customers access the application in the last fifteen seconds. (Any number could access the system in between these two time intervals.)

Conditional Probability

1. In a class of 25 students 14 like Pizza and 16 like coffee. One student likes neither and 6

students like both. One student is selected from the class. What is the probability that the

student

a. Likes pizza 14/25

b. Likes pizza given that he/she likes coffee? 6/16

Additive and Multiplicative Rules

2. If P(X) = .5 and P(Y ) = .7 and X and Y are independent determine the probability of

the occurrence of:

a. Both X and Y b. X or Y c. X given that Y occurs

3. If p(X) = 0.23 and p(X ∩ Y ) = 0.12 and P(X ∪ Y ) = 0.34 then P(y′) =

1. .23 b. .52 c. .11 d. .77 e. .48

4. A survey of families revealed that 8% of all families eat turkey at holiday meals, 44% eat

ham, and 16% have both turkey and ham.

a. What is the probability that a family selected at random had neither turkey nor ham at

their holiday meal?

a. P(neither ham nor turkey) = 1 - P(ham ∪ turkey)

= 1 - [P(ham) + P(turkey) - P(ham ∩ turkey)]

b. What is the probability that a family selected at random had only ham without having

turkey at their holiday meal? P(ham only) = P(ham) - P(ham ∩ turkey) =

c. What is the probability that a randomly selected family had ham at their holiday meal,

given that they had turkey?

P(ham turkey) =

P(ham ∩ turkey)/P(turkey)

=

9. How many different 6−letter arrangements can be formed using the letters in the word

ABSENT, if each letter is used only once?

a. 6 b. 36 c. 46656 d. 720

10. Bolts produced by a machine vary in quality. The probability that a given bolt is defective is 0.03. A random sample of 35 bolts is taken from the week’s production. If X denotes the number of defectives in the sample, find the mean and standard deviation of X

.

11. Of pre-med students in a private university, on average only 36% of students enrolled in

a given section of organic chemistry will pass. What is the probability that Sarah will have

to take the class three times in order to pass?

P=0.36

K=3

17. An insurance company pays hospital claims. The number of claims that include

emergency room or operating room charges is 85% of the total number of claims.

The number of claims that do not include emergency room charges is 25% of the total number of claims. The occurrence of emergency room charges is independent of the occurrence of operating room charges on hospital claims.

Calculate the probability that a claim submitted to the insurance company includes

operating room charges.

(A) 0.10 (B) 0.20 (C) 0.25 (D) 0.40 (E) 0.80

20 An insurance company issues life insurance policies in three separate categories:

standard, preferred, and ultra-preferred. Of the company’s policyholders, 50% are

standard, 40% are preferred, and 10% are ultra-preferred. Each standard policyholder

has probability 0.010 of dying in the next year, each preferred policyholder has

probability 0.005 of dying in the next year, and each ultra-preferred policyholder

has probability 0.001 of dying in the next year.

A policyholder dies in the next year.

What is the probability that the deceased policyholder was ultra-preferred?

(A) 0.0001 (B) 0.0010 (C) 0.0071 D) 0.0141 (E) 0.2817

30. An actuary has discovered that policyholders are three times as likely to file two claims

as to file four claims.

If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?

46. A device that continuously measures and records seismic activity is placed in a remote

region. The time, T, to failure of this device is exponentially distributed with mean

3 years. Since the device will not be monitored during its first two years of service, the

time to discovery of its failure is X = max(T, 2) .

Determine E[X].