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Probability: Continuous Random Variables

1.) Suppose we are producing copper wire and putting the wire on spools. Each spool contains 100 feet of wire. Defects such as nicks in the wire can occur at random locations. What would be a reasonble distribution for each of the following: (a) the number of spools produced until a spool is produced that contains one or more defects?
(b) the number of defects on the next spool of wire?
(c) whether or not the first defect is within the first 100 feet of wire on the next spool?
(d) the number of spools out of the next 25 that are free of defects?
(e) the length of wire produced until the first defect?
(f) the location of the first defect on a spool of wire given that the spool contains exactly one defect

2.) Let Y be a Poisson random variable with parameter 2.5
(a) Compute Pr{Y=0}
(b) Compute Pr{Y <= 1} ("<=" means less than and equal to)
(c) Compute Pr{Y=0 | Y <= 1}
(d)What is E[y]?
(e) Determine all medians of Y
(f) What is the probability that Y is an odd integer?

3.) Let X be a continuous random variable with probability density function f(s) = c/(1+ (s^2)) for
-2 <= s <= 2.
(a) Determine c.
(b) Determine Pr{X <= 0}.
(c) Determine the mean of X
(d) determine all medians of X
(e) Compute Pr{X=2 | X >= 0}
(f) Determine the cumulative distribution function
* the cumulative distribution function is :
F(t)= P{X <= t} = 1 - e^ (-ut) , where t> 0

4.) Let Y be an exponential random variable with parameter 2.5
(a) Computer Pr{Y=0}
(b) Compute Pr{Y > x+2 | Y > x} for x >= 0?
(c) What is E[Y]?
(d) determine all medians of Y
(e) what is the probability that Y is an odd integer
(f) Determine the cumulative distribution function of Y.

* Can you please give a detailed explanation as to how you solve these kinds of problems...I have other problems such as the ones i typed that i need to do as well. It will really help if you can show how you solve the problems. thanks.

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Solution Preview

1.(a)Let p be the probability that a spool has atleast 1 defect.
Chance that kth spool had defect = chance that first k-1 have no defects X kth has defect= (1-p)^k-1*p

This is a geometric random variable.

(b)Let the average defect rate be x nicks per feet.You can get the average by using a very large number of observations.Then the number of nicks in a 100 feet wire is a poisson variable with parameter 100*x.

(c)Let X be the variable.
X=0 with chance of no defect in first 100 feet.
X=1 (1 - chance of no defect within first hundred feet).

X is a bernoulli random variable

(d)Let Y be the variable
p is the chance that a spool is not defective
chance that Y =k is
25Ck (25 choose k)*p^k(1-p)^25-k
this is a binomial variable

(e)Divide the wire into k infinitesimall chunks.....chance of defect
in a chunk be p.Also p->0 and k->infinity
(Chance that length >=kp) = (1-p)^k....
=((1-p)^p)^(k*p) = (e-(k*p))
exponential distribution. kp=constant ......

(f)It is a uniform random variable.The defect is equally likely
to be any where by symmetry.....if u break the wire into n parts
such that n is very large and let p be the probability that any these parts has the defect..then chance that any part will have it and rest of the parts wont have it is ...

Solution Summary

Probability problems involving random variables are solved.