State whether the following statements are True or False and also give the reason in support of your answer.

(a) If odds in favour of an event A are 2:5 then probability of happening of the event A will be .

(b) If two independent events A and B are such that A and B are independent. P(A) = and P(B) =  , then events A and are independent.

c) If X is a discrete random variable with probability mass function (pmf)

 Ms. X thinks that there is a 30% chance that her company will set up a branch office in New Delhi. If it is so, then she is 60% certain that she will be assigned the responsibilities of a branch manager there. What is the probability that Ms. X will be branch manager at the New Delhi office?

 A company has installed 10,000 electric lamps in a metro. If these lamps have an average life of 1,000 burning hours with a S.D. of 200 hours. Assuming normality, what number of lamps might be expected to fail. 

i)  in the first 800 burning hours. 

ii) between 800 and 1200 burning hours.        

iii) more than 1000 burning hours  

The distribution function F of a continuous variable is given by

F(x) = 0, if x < 0

= x 2 , if 0 < x < 1

= 1, if x > 3

(i) Find the pdf of X,

(ii) Evaluate P[|X| <1/ 2], using distribution function F,

(iii) Evaluate P[|X| < 1/ 2], using pdf.

The joint density function of random variables X and Y is given by  

Are X and Y independent?
 

 The metro trains on a certain section run every 10 minutes between 5 a.m. to 10 p.m. What is the probability that a commuter entering the station at a random time during this period will have to wait  

(i) at least five minutes?

(ii) zero minute? 

(iii) 2 to 5 minutes?

 A smoke-detector system uses two devices A and B. If smoke is present the probability that it will be detected by device A is 0.95; by device B, 0.98 and by both devices, 0.94. If smoke is present find the probability that it will be detected