Section
|
Number
|
Point
|
Comments
|
6.5
|
4
|
3
|
|
|
6
|
4
|
|
|
18
|
4
|
|
|
20
|
4
|
|
|
34
|
4
|
Write this up carefully
|
|
47
|
3
|
|
adhoc
|
A
|
4
|
Show that the variance of the
Poisson distribution with parameter lambda is also lambda. You'll find
it helpful to compute E(X^2) by
writing k^2=k(k-1) +k.
|
|
B
|
3
|
Show that for any real number
"a",
Var(aX)=a^2*Var(X).
|
|
C
|
3
|
Compute the mean and the
variance of the normalized binomials that were used in the CLT example
(slide #1 in "probability6.pdf").
|
|
D
|
6
|
You are trying to estimate the
probability "p" that a coin lands "H". How large should n be so that
your empirical average (#of H in n flips / n) is within 0.01 to "p"
with probability of at least 0.9? You will found the following useful:
Chebyshev's inequality, the maximum of p*(1-p) is <= 1/4.
|
|
E
|
2
|
From past experience a professor
knows that the test score of a student taking the final exam is a
random variable with mean 75. Give an upper bound for the probability
that a student's test score will exceed 85.
|
|
F
|
2
|
Suppose that in addition the
professor knows the variance of a student's test score is equal to 25.
What can be said about the probability that a student will score
between 65 and 85?
|