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

Are X and Y independent?
 

The probability that a person living in a certain city owns a dog is estimated to be 0.3. Find the probability that the tenth person randomly interviewed in that city is the fifth one to own a dog.

 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

The daily consumption of milk in a city, in excess of 20,000 litres, is approximately distributed as a gamma variate with parameters r = 2 and λ = 1/10,000. The city has a daily stock of 30,000 litres. What is the probability that the stock is insufficient on a particular day?

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)

 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