Tutorial-3 and Tutorial-4 Binomial Distribution 1. Forty percent of business travelers carry either a cell phone or a laptop. In a sample of 15

business travelers. (a) What is the probability that three have a cell phone or laptop? (b) What is the probability that 12 of the travelers have neither a cell phone nor a

laptop? (c) What is the probability that at least three of the travelers have a cell phone or a

laptop? 2. A university found that 20% of its students drop out without completing the introductory

statistics course. Assume that 20 students have registered for the course this quarter. (a) What is the probability that two or fewer will dropout? (b) What is the probability that exactly four will dropout? (c) What is the probability that more than three will dropout? (d) What is the expected number of withdrawals?

3. A satellite system consists of 4 components and can function adequately if at least 2 of the 4 components are in working condition. If each component is, independently in working condition with probability 0.6, what is the probability that the system functions adequately?

4. The phone lines to an airline reservation system are occupied 50% of the time. Assume that the events that the lines are occupied on successive calls are independent. If 10 calls are placed to the airline, What is the probability that

(a) for exactly three calls the lines are occupied? (b) for at least one call the lines are not occupied? (c) what is the expected number of calls in which the lines are all occupied? 5. With the usual notations, find p for a binomial random variable x, if n= 6 and if

9 P (X=1) = P(X=2). 6. The mean and variance of a binomial distribution are 3 and 2 respectively. Find the probability that the variate takes values. (i) Less than or equal to 2. (ii) Greater than or equal to 7. Poisson Distribution 1. The number of monthly breakdowns of computer is a random variable having Poisson

distribution with mean equal to 1.8. Find the probability that this computer will function for a month

(a) without a breakdown (b) with only one breakdown (c) with atleast one breakdown 2. It is known that the probability of an item produced by a certain machine will be

defective is 0.05. It the produced item are sent to the market in packets of 20, find the number of packets containing at least, exactly and at most 2 defective items in a consignment of 1000 packets using (i) binomial distribution and (ii) Poisson approximation to binomial distribution.

3. It the numbers of telephone calls coming into a telephone exchange between 9.00 a.m. and 10.00 a.m. and between 10.00 a.m. and 11.00 a.m. are independent and follows Poisson distributions with parameter 2 and 6 respectively. What is the probability that more than 5 calls come between 9 a.m. and 11 a.m.?

4. If X and Y are independent Poisson vairates such that P[X=1]=P[X=2] and P[Y=2]=P[Y=3]. Find (i) the variance of X-2Y (ii) P[X=0] (iii) P[0

Geometric Distribution 1. Suppose a random variable X has a geometric distribution with a mean of 2.5. Determine

the following probabilities: (a) P(X=1) (b) P(X4) 2. If the probability that an applicant for drivers license will pass the road test on any given

trial is 0.8. What is the probability that he will finally pass the test (a) on the fourth trial (b) in fewer than 4 trails?

3. A person decides to toss a coin until first head occurs. If the probability of obtaining head is 0.25. What is the expected number of tosses to get the first head?

4. A lot of TV tube is tested randomly. Testing is done one after another until a defective tube is found. The number of tubes tested determine whether the lot is accepted or not. If k be the number such that, the lot is accepted if k or more tubes are tested for the first defective. Then (i) if k=5 and p=0.1, find the probability of accepting the lot, (ii) what would be the value of k if we want to be 90% confident of rejecting a lot that has 25% defective?

5. Identify the distribution with the following M.G.F: M.G.F. (5-4et)-1. Find p[X>5] and mean and variance of the distribution.

Hyper geometric distribution 1. Let a random variable X has probability mass function,

{ } ( ) ( )

. 0 integers. positive aren and MN, re Whe

,min,0max ,Pr

otherwise

nMxNnM

nN

xnMN

xM

xX

=

+

==

(i) Construct the graph of probability mass function when (a) N = 6, M =4, n = 3. (b) N= 6, M = 2, n = 4.

(ii) Compute the mean of X and Variance of X in part (i). (iii)Compute the following probabilities,

(a) { }2Pr X (b) { }1Pr X and (c) { }20Pr X using parameters given in part (i). 2. Consider a lot consisting of 50 bulbs is inspected by taking at random 10 bulbs and testing

them for defective or non-defective. If the number of defective bulbs in a lot containing 50 bulbs is 10, compute the following probabilities (i) Pr{The sample of 10 bulbs contains 2 defective bulbs} (ii) Pr{The sample of 10 bulbs contains at least 2 defective bulbs} (iii) Pr{The sample of 10 bulbs contains at most 3 defective bulbs} Further, if the sample of 10 bulbs contains at most 1 defective bulbs, the lot containing 50 bulbs is accepted, otherwise it is rejected. Compute the probability that the lot containing 50 bulbs is accepted.

3. Consider a lot consisting of 1000 bulbs is inspected by taking at random 10 bulbs and testing them for defective or non-defective. If the proportion of defective bulbs in a lot containing 1000 bulbs is 0.06, compute the probabilities stated in example 2. State the approximation used in computing these probabilities.

4. Use Q. 5 of PD. Use HGD to answer the question. 5. The environment protection agency has purchased 40 precision instruments to be used to

measure the air pollution at various locations. 8 of these are randomly selected and tested for defects. If 4 of the 40 instruments are defective, what is the probability that the sample will contain not more than one defective? Use BD approximation to find the same probability and compare. Also the PD approximation to BD and get the results.

Answers Binomial distribution 1. Let X: travelers having cell phone or laptop. Then ).,(~ 4015BDX

(a) P(X=3)= 123 443

15)(.)(.

(b) P(X=12)= 312 461215

)(.)(.

(c) P(X>=1)=1-P(X=0)

2. Let X: Drop out students. Then ).,(~ 2020BDX .

(a) P(X3)=1-[P(X=0)+P(X=1)+P(X=2)] (d) E(X)=4 3. Let X: number of components. Then ).,(~ 604BDX .

The required probability is P(X>=2)=P(X=2)+P(X=3)+P(X=4) 4. Let X: lines are occupied. Then ).,(~ 5010BDX

(a) 73 553

103 )(.)(.)(

==XP (b) )()( 011 == XPXP (c) E(X)=5

5. p=18/23 6. Given np=3 and np(1-p)=2. Solving we get p=1/3 and n=9. Now (i) and (ii) can be easily

found. Poisson distribution 1. Let X: number of monthly breakdowns and =1.8.

(a) P(X=0)= 81.e , (b) P(X=1)= 8181 .*. e ((c) P(X>=1)=1-P(X=0) 2. Let X:the item is defective. Then given p=0.05 and n=20. Now (i) According to BD, the number of defective items are 1000*P(X>=2), 1000*P(X=2)

and 1000*P(X=2), 1000*P(X=2) and 1000*P(X5). 4. Solving P(X=1)=P(X=2), we get =2 and solving P(Y=2)=P(Y=3), we get =3. Therefore,

)(~ 2PDX and )(~ 3PDY . Therefore, )(~ 5PDYX + . Now (i) V(X-2Y)=V(X)+4V(Y)-2COV(X,Y) = 14

5. p = 20/100 = 0.2, n=10, Required probability = P(X>=9) = P(X=9)+ P(X=10) Geometric distribution 1.Given q/p=2.5. Then p=1/3.5. Then (a), (b), (c), and (d) follows 2. Let X: pass the test.Then p=0.8, (a) P(X=4)=(.8)(.2)3 and (b) P(X=5)=0.6561 (ii) p=0.25 and P(X5)=1-P(X

Hypergeometric distribution 1. (ii) (a) E(X)=0.5, V(X)=0.4, (b) E(X)=1.33, V(X)=0.355 (iii) (a) { }2Pr X =1- { }1XPr , (c) { } )()()(Pr 21020 =+=+== XPXPXPX

2. Given N=50, M=10, n=10. (i) P(X=2)=

1050

840

210

/ , (ii) { }2XPr , (iii) { }3XPr . The

lot is accepted if { }1XPr .

3. Use BD assumption. P=0.06, (i) P(X=2)= 82 94062

10)(.)(.

, Similarly (ii) { }2XPr ,

(iii) { }3XPr . The lot is accepted if { }1XPr . 4. Given N=100, M=20, n=10. To find

P(X>=10)=P(X=9)+P(x=10)=

+

10100

1020

10100

180

920

//

5. Given N=40, M=4, n=8. To find P(X