Hello tutors, can I please get help with the following ? Thank you.

1) A machine has n identical components each of which has an exponentially distributed lifetime X with expected value 10. The total life of the n components is also a random variable, since it is a function of random variables. We denote that total life as T.

What is the expected value and variance of the total life (T) of all n components together, respectively?

The probability of any interval of T will ...

If Y=10X, what is the variance of Y equal to?

Answer choices:

- will be very accurately approximated by a Gaussian distribution

- be constant

- E(T) = 10n

- Var(T) = 100n

- not be very accurately approximated a Gaussian distribution

- an exponential distribution

- 10000

2)Consider the experiment of drawing a random sample of 100 bags of fertilizer to see whether a factory producing them keeps good quality control. The bags specification says that the net wait of a bag is 50 pounds. Of course, we all know that the net weight put by factories in products is the average weight. A little departure from the average here and there is ok, but if the probability that there is a lot of departure from that weight is high, quality control engineers know that that is a bad sign. Something is wrong with the production process in that case.

Let X denote the actual net weight of a bag of fertilizer. The average (expected) weight of a bag of fertilizer is 50 pounds and the variance is 1.

a) TheE(X) :

b)Thex equals :

c) 50.25 pounds is how many standard deviations above the expected value for E(X)?

d) The probability that the average weight is between 49.75 and 50.25 is approximately equal to:

answer choices for a), b), c), and d):

-0.9876

-5000

-15

-0.1974

-0.5

-0.1

-0.0062

-2.5

-50

-1