Probability and Statistics IISER Pune January 2015

Assignment 3, Page 1 of 3 January 2015

1. Assuming all sex distributions to be equally probable, what proportion of familieswith exactly six children should be expected to have three boys and three girls?

Ans: Let X denote the number of boys in a family of 6 children.P (X = 3) =

(63

)(12)6 = 5

16= 31.25%

2. If the probability of hitting a target is 15

and ten shots are fired independently, what isthe probability of the target being hit at least twice? Find the conditional probabilitythat the target is hit at least twice, assuming that at least one hit is scored.

Answer: Let X denote the number of successful shots. X Bin(10, 15).

P (X 2) = 1 (P = 0) P (X = 1) = 1 (45)10 10 1

5(45)9.

P (X 2 | X 1) = P (X2)P (X1) =

1( 45)1010 1

5( 45)9

1( 45)10

3. A lot consisting of 100 bulbs is inspected by taking at random 10 bulbs and testingthem. If the number of defective bulbs is at most 2, the lot is accepted, otherwise itis rejected. If there are in fact 10 defective bulbs in the lot, what is the probabilityof accepting the lot.

And: P (Accepting the lot ) =2

x=0

(10x )(90

10x)(10010 )

4. A book of 500 pages contains 500 misprints. Estimate the chances that a given pagecontains at least three misprints.

Ans: X=number of misprints on given page. X Poisson(1). P (X 3) = 0.08.5. Two people toss a true coin n times each. Find the probability that they will score

the same number of heads.

Ans: X=number of heads by the first person. Y=number of head by the secondperson.

P (X = Y ) =n

k=0 P (X = k, Y = k) =n

k=0 P (X = k)P (Y = k) =n

k=0

(nk

)2(12)2n

6. In rolling six true dice, find the probability of obtaining (a) at least one, (b) exactlyone, (c) exactly two, aces. Compare with poisson approximations.

Ans: Success(S)=getting an ace; Failure(F)=not getting an ace. P (S) = 16, P (F ) =

56. Random experiment here is a sequence of 6 independent Bernoulli trials with

probability of success 16. X = the number of successes in 6 independent Bernoulli

trails. X Bin(6, 16). (a) P (X 1) = .6651, (b) P (X = 1) = .4018, (b) P (X =

2) = .2009Poisson Approx. X Poisson(1). (a) 1 e1 = .6321, (b) .3679, (c) 0.1839

7. If there are on the average 1 per cent left-handers, estimate the chances of having atleast four left-handers among 200 people. And: X=number of left-handers among200 people. E(X) = = 2. X Poisson(2). P (X 4) = 0.143.

8. A burnt out bulb was mistakenly placed in a box containing 3 good bulbs. In orderto locate the bad bulb, bulbs are randomly tested one by one, without replacement.Let X denote the number of bulbs tested to determine the bad bulb. Find the pmfand the cdf of X.

Ans: P (X = i) = 14, for i = 1, 2, 3, 4.

9. An animal either dies (D) or survives (S) in the course of a surgical experiment. Theexperiment is to be performed first with two animals. If both survive, no further trailsare to be made. If exactly one animal survives, one more animal is to undergo theexperiment. If both animals die, two additional animals are to be tried. Assumingthat the trails are independent and that the probability of survival in each trial is2/3, find the probability distribution of the number of survivals and the number ofdeaths.

10. Suppose that the distribution function of X is given by

F (x) =

0 x < 0x4

0 x < 112

+ x14

1 x < 21112

2 x < 31 3 x

(a) Find P (X = i), i = 1, 2, 3. (b) Find P{12< X < 3

2}.

11. If the distribution function of X is given by

F (x) =

0 x < 012

0 x < 135

1 x < 245

2 x < 3910

3 x < 3.51 3.5 x

Find the probability mass function (pmf) of X, E(X) and V (X).

12. Let

F (x) =

0 x < 0

x2 0 x < 12

1 3(1 x)2 12 x < 1

1 x 1Verify that F is a cumulative distribution fuction of some random variable X.Determine P (1

4< X < 3

4), E(X) and V (X).

13. A state lottery has one million tickets each costing Rs. 10. There is one first prize ofRs. 50,000; nine second prizes of Rs. 2,500 each; ninety third prizes of Rs. 250 eachand 900 prizes of Rs. 25 each. Find the expected profit if you buy 5 tickets. Whatis the expected profit to the state if only 80% of the tickets are sold?

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

f(x) =

x+141 x 1

3x4

1 x 30 otherwise

Find the cumulative distribution function, mean and variable of X. Also evaluateP(1

2< X 5

2

)and P

(| x |< 32

).