1. ## Question about Exponential Random Variables.

Suppose that the time T until a type of television set fails is an exponential r.v. with rate 0.2. Recall the probability density function of T, and the cumulative distribution of T.

(a) Find the probability that failure occurs between 3 and 5 years.

(b) If a TV set did not fail in 10 years, what is the chance that it will fail in the next year?

(c) A school buys 10 TV sets. Let N be the number that fail during the first 4 years. Find the distribution of N.

All help greatly appreciated!

2. I'll give you a few tips:

like the question say, you first need to write yourself the density function and the cumulative distribution function.

a.
it T is time in years, and T~Exp(0.2), then you are looking for P(3<T<5). There are 2 ways to do it, one is the use the cumulative distribution function, and the other is to use the density function.
P(3<T<5) = P(T<5) - P(T<3) , while P(T<5) and P(T<3) can be obtained from the CDF. Another way, is to calculate an Integral of the density function using the values 3 and 5.

b.
at first sight looks like a conditional probability, but look on the internet for the "Memorylessness" attribute of exponential distribution.

c.
define a "success" or "failure" to be a failure during the first 4 years. Calculate the probability of that to happen in a random TV set. Then you will have an experiment of 10 TV sets, while you know the probability p of each one of the 10 to "succeed" or "fail". Then what should be the distribution of N then ? ( Hint: It's discrete )