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Hey all

I am having trouble with statistics, and I am wondering if any of you can help me out with these few questions with steps. I cannot seem to do them properly. Wondering if any statistic wizards can guide me through

1. A system consists of 3 components arranged in series. The lifetime (in
days) of each component follows approximately an exponential distribution with
a mean lifetime of 100 days. The lifetimes of the components are independent.

(a) What is the probability that the first component lasts between 50 and 100
days?

(b) What is the probability that 2 of the 3 components have lifetimes between 50 and 150 days?

(c) What is the cumulative distribution function of the lifetime of the entire system? What are the corresponding median and mean lifetime?

(d) If the 3 components are arranged in parallel, what is the c.d.f. of the lifetime of the entire system? What is the corresponding median lifetime?


2. Suppose that X is a normal random variable with mean 3 and variance 4.

(a) Calculate P(X > 4),
(b) Calculate P(2 < X < 4),
(c) Calculate P(|X − 3| < 2),
(d) How does the the probability in (c) compare with P(|X − 2| < 2) (larger / equal / smaller)? Why?
(e) Find c such that P(|X − 3| < c) = 0.90


3. A machine fills 25-pound bags of dry concrete mix. The actual weight
of the mix that is put in the bag is a normal random variable with standard
deviation at 0.05μ pound. The mean μ can be set by the machine operator.

(a) At what mean weight should the machine be set so that at most 10 per
cent of the bags are underweight?

(b) Find the corresponding mean weight to be set for larger 50-pound bags.


Thank you for all those who helped