Skeleton Answers
Midterm, CSC 354S, St. George Version
Spring 2000

1. [loaded die]
a)
xi	1	2	3	4	5	6
p(xi)	1/21	2/21	3/21	4/21	5/21	6/21
A pmf graph will have lines of height p(xi) at x-axis points xi.

x	(-inf,0)	[1,2)	[2,3)	[3,4)	[4,5)	[5,6)	[6,+inf)
F(x)	0		1/21	3/21	6/21	10/21	15,21	21/21
A cdf graph will be a step function, rising at each xi {1,...6} to the height F(xi).

b)
E(x) = 1(1/21) + 2(2/21) + ... + 6(6/21) = 91/21 = 4.33
E(X**2) = 1**2(1/21) + 2**2(2/21) + ... + 6**2(6/21) = 21
Var(X) = 21 - (91/21)**2 = 21 - 18.78 = 2.22

2. [lamp]
prob that the lamp will last longer than its mean life of 3000 hours is
P(X>3) = 1 - P(X <= 3) = 1 - F(3)
For expo(beta), F(x) = 1 - e**(-x/beta), x >= 0. So P(X>3) = 1 - (1 -e**(-3/3)) = 
e**(-1) = 0.368
[So regardless of the value of beta, the result will always be the same !!!]

3. [MLE]
L(a) = 1/((-a)**n) and we must have a <= Xi <= 0 for all i. Thus, L(a) is maximized by setting a to be as large as possible, i.e., as close to 0 as possible as a < 0. This gives
a hat = X sub (1), the first order statistic or the smallest Xi.

4. and 5. deal with topics not covered yet.