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               Econ 120A, Winter 1998  --  Homework #2 (5%)
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This homework is due at 11:10 am on Tuesday, February 24, 1998.  Papers
turned in at the end of the class will carry a 10% penalty and those
turned in later than that will carry a 25% penalty PER DAY.  As before,
team up with others (maximum 3 persons per team) and submit a single
paper with all names.

I.
A continuous random variable X has the density function

                                   -(x/a)
    f(x) = (1/a)exp(-x/a) = (1/a) e             x >= 0

where a is a fixed positive constant, and exp is the exponential 
function.  

1.  Derive expressions for P(X >= 2) and P(X >= 4 | X >= 2), where
    | means "given".  
2.  Suppose X stands for the life of a fluorescent light bulb (in
    hours) with a = 400.  Compute the probability that a light bulb
    chosen at random will have life at least 500 hours.
3.  Use the probability obtained in I.2 to calculate the probability
    that 4 out of 5 light bulbs will have life at least 500 hours.

II.
The life, in hours, of a battery is normally distributed as N(200, s2),
where s2 is sigma squared the variance, an unknown.  The manufacturer
requires that at least 90 percent of the batteries should have lives
exceeding 150 hours.  Calculate how large sigma can be so that the
above condition is satisfied.

III.
The distribution of personal incomes of families has been found to have 
the following probability density function.

                 -(a+1)
     f(x)  =  a x              a >= 5   and  1 <= x < infinity

that is, a times x to the power -(a+1).  f(x) = 0 elsewhere.

1.  First integrate f(x) from 1 to infinity and show that it is 1.
2.  Next compute E(x), E(x squared), E(x cubed), and E(x to the fourth 
    power in terms of a.
3.  Suppose y = x squared.  Compute Var(x), Var(y), and Cov(x,y) in 
    terms of a.
4.  Assume that a = 5 and compute the numerical value of the coefficient
    of correlation between x and y.  Note that even though x and y have
    an exact relationship, the coefficient of correlation is not 1.