Assignment:
Q1.) 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
Q2.) 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?
Q3.) 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
Q4.) 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.
Provide complete and step by step solution for the question and show calculations and use formulas.