1 Sundry
Before you start your homework, write down your team. Who else did you work with on this
homework? List names and email addresses. (In case of homework party, you can also just describe
the group.) How did yo
...
1 Sundry
Before you start your homework, write down your team. Who else did you work with on this
homework? List names and email addresses. (In case of homework party, you can also just describe
the group.) How did you work on this homework? Working in groups of 3-5 will earn credit for
your "Sundry" grade.
Please copy the following statement and sign next to it:
I certify that all solutions are entirely in my words and that I have not looked at another student’s
solutions. I have credited all external sources in this write up.
I certify that all solutions are entirely in my words and that I have not looked at another student’s
solutions. I have credited all external sources in this write up. (Signature here)
2 How Many Kings?
Suppose that you draw 3 cards from a standard deck without replacement. Let X denote the number
of kings you draw.
(a) What is Pr(X = 0)?
(b) What is Pr(X = 1)?
(c) What is Pr(X = 2)?
(d) What is Pr(X = 3)?
(e) Do the answers you computed in parts (a) through (d) add up to 1, as expected?
(f) Compute E(X) from the definition of expectation.
(g) Suppose we define indicators Xi, 1 i 3, where Xi is the indicator variable that equals 1 if
the ith card is a king and 0 otherwise. Compute E(X).
(h) Are the Xi indicators independent? How does this affect your answer to part (g)?
3 Sisters
Consider a family with n children, each with a 50% chance of being male or female. Let X be the
total number of sisters that the male children have, and let Y be the total number of sisters that the
female children have (for example, if n = 3 and there are two boys and one girl, then X = 2 and
Y = 0). Find expressions for E(X) and E(Y) in terms of n. Do we expect that boys have more
sisters or girls have more sisters?
[Hint: Define a random variable B to denote the number of boys, find an expression for X as a
function of B, and apply linearity of expectation. Use a similar approach for girls.]
4 Unbiased Variance Estimation
We have a random variable X and want to estimate its variance, s2 and mean, µ, by sampling from
it. In this problem, we will derive an “unbiased estimator” for the variance.
(a) We define a random variable Y that corresponds to drawing n values from the distribution for
X and averaging, or Y = (X1 +...+Xn)/n. What is E(Y)? Note that if E(Y) = E(X) then Y is
an unbiased estimator of µ = E(X). Hint: This should not be difficult.
(b) Now let’s assume the actual mean is 0 as variance doesn’t change when one shifts the mean.
Before attempting to define an estimator for variance, show that E(Y 2) = s2/n.
(c) In practice, we don’t know the mean of X so following part (a), we estimate it as Y. With this
in mind, we consider the random variable Z = Ân i=1(Xi "Y)2. What is E(Z)?
(d) What is a good unbiased estimator for the Var(X)?
(e) How does this differ from what you might expect? Why? (Just tell us your intuition here, it is
all good!)
5 Markov Bound for Coupon Collectors
Suppose you are trying to collect a set of n different baseball cards. You get the cards by buying
boxes of cereal: each box contains exactly one card, and it is equally likely to be any of the n cards.
You are interested in finding m, a lower bound on the number of boxes you should buy to ensure
that the probability of you collecting all n cards is at least 1 2. In class, we used the Union Bound to
show that it suffices to have m $ nln(2n).
Use Markov’s Inequality to find a different (weaker) lower bound on m.
Soluti
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