BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI- K. K. BIRLA GOACAMPUS

First Semester 2015-2016Tutorial Sheet - 4

Course No. MATH F113 Course title: Probability and StatisticsDate: sep 03, 2015

1. Verify that the following functions are probability density functions and find CDF:

(i) f(x) =

{|x|, |x| < 10, elsewhere

(ii) f(x) =

12x2, 0 < x 1x2 + 3(x 1)2, 1 < x 2x2 + 3(x 1)2 + 2(x 2)2, 2 < x 30, elsewhere

2. Proposed cumulative distributions are given, find the density that would be associated witheach, and decide weather it really does define a valid continuous density. If it does not,explain what property fails

(i) F (x) =

0, x < 1x+ 1, 1 x 01, x > 0

(ii) F (x) =

0, x 0x2, 0 < x 1

2x2, 1

2< x 1

1, x > 1

3. If X is uniformly distributed with mean 1 and variance 43, find P (X < 0) .

4. If X U(0, 1), let Y = a+ (b a)x, then Y U(a, b)5. If X has a uniform distribution in [0, 1], find the distribution of 2 logX. Identify the

distribution also.

6. Subway trains on a certain line runs every half hour between midnight and six in themorning. What is the probability that a man entering the station at a random time duringthis period will have to wait at least twenty minutes?

7. Show that the graph of the density for a Gamma random variable with parameter and assumes its maximum value at x = ( 1) for > 1.

8. (a)Prove the following:

(i) If Xi (i, ), i = 1, 2, ..., n are independent thenn

i=1Xi (n

i=1 i, ).

(ii) If X (, ), then for any k > 0, kX (, k)(iii) If Xi, i = 1, 2, ..., n are independent exponentially distributed random variables with thesame parameter , then

ni=1Xi has the Gamma distribution with parameters n and .

(iv)If the inter-occurrence times of events are independent exponential variates with the sameparameter , then the time until the rth occurrence of the event has Gamma distribution withparameters r and .

(v) If the inter-occurrence times of events are independent exponential variates with the sameparameter , then Xt, the number of events occurring in any interval of length, t say [0, t) hasPoisson distribution with parameter t, Where = 1

.

9. Instantaneous surges of electrical current occur randomly and independently on a particularline at an expected rate 0.1 per hour. The electrical system will fail after 4 surges. Determinethe probability that the system is still functioning in 10 hours.

10. Calls to an internet service provider are made independently and random at a constantexpected rate during the early morning hrs. It is currently 2 A. M. and the service providerslines are all open. What is the probability that all the lines will remain open for the next 5minutes, if the source provider expects 10 calls over the next 1

2hour ?

11. A professor takes early retirement on July 1st exactly 3 months after his 60th birthday andpurchases a term life insurance policy for 25,000 dollars, the maturity date coincides with the65th birthday. According to a mortality table for similarly situated men, the probability thata male of exact age 60 dies before reaching his 61st birthday is 0.0138. Determine theprobability that the professor dies before reaching his 61st birthday if deaths are assumed tobe exponentially distributed between these days.

12. The life length X of an electrical component follows and exponential distribution. There are 2processes by which component may be manufactured. The expected life length of thecomponent is 100 hrs. if process I is used to manufacture, while 150 hrs. if process II is used.The cost of manufacturing a single component by process I is Rs. 10 /- while it is Rs. 20 /- forprocess II. Moreover, if the components last less than the guaranteed life of 200 hrs., a loss ofRs. 50/- is to borne by the manufacturer. Which process is advantageous to manufacturer ?

13. Let X be a random variable with E(Xm) = (m+ 1)!2m,m = 1, 2, 3, ..... Find the density of X.

14. Prove the memoryless property for exponential distribution.

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