probability2

 

 typed answers with explanation either in doc or in pdf. 

 

 strict due. 9/18 by midnight in EST

 

 for 7, you will need to use R program http://www.r-project.org

 

 for (i in 1:20) {

dice <- sample(1:6,10,replace=TRUE); print(dice); print(sum(dice)); }

Math 464 – Fall

1

1 – Homework 3

1. The probability mass function of a discrete RV X is given in the table
below. Compute the following:

(a) the probability X is eve

n

(b) the probability that 1 ≤ X ≤ 8
(c) the probability that X is −4 given that X ≤ 0.
(d) the probability that X ≥ 3 given that X > 0.

x -4 -1 0

2

4 5 6
f(x) 0.15 0.2 0.10 0.1 0.2 0.2 0.05

2. Compute the mean and variance of the geometric distribution.
Hints: It is sometimes easier to compute E[X(X −1)] than E[X2]. Note that
E[X(X − 1)] = E[X2] − E[X]. The geometric series formula is

∞

∑

n=0

r
n =

1

1 − r

If you differentiate this once and then again you get the useful identities:

∞
∑

n=1

nr
n−1 =

d

dr

1

1 − r
,

∞
∑

n=2

n(n − 1)rn−2 =
d2

dr2
1

1 − r

The following is not to be turned in. This same “summation by differen-
tiating” trick can be used in other setting. Take the binomial identity

(x + y)n =
n

∑

k=0

(

n

k

)

x
k
y
n−k

Differentiate with respect to x, then set x = p, y = 1 − p and you get
an identity that helps you compute the mean of the binomial distribution.
Differentiate twice and you get an identity that will lead to the variance.

3. (Exposition) Let X be a discrete RV whose range is 0, 1, 2, 3, · · ·. Prove
that

E[X] =
∞
∑

k=0

P(X > k)

1

4. (Exposition) Let X be the number of emails that a company receives
in a day. Assume that X is a Poisson random variable with parameter λ.
The company classifies each email as spam or not spam. The probabilty that
a single email is spam is p. Let Y be the number of spam emails that the
company receives in a day. Assume that for any two emails, whether or not
they are spam are independent events.
(a) Find E[Y ].
(b) Show that Y is also a Poisson random variable and find its parameter (in
terms of λ and p). Hint: Use the partition theorem:

P(Y = k) =
∞
∑

n=k

P(Y = k|X = n)P(X = n)

If we are given that X = n, then Y is a binomial RV.

5. A coin has probability p of heads. We flip it until we get 3 heads in a row.
Let X be the total number of flips. Find the expected value of X. Hint: The
idea is to use the partition theorem and the fact that whenever you get a T,
things start over. In class we looked at the probability of a run of 3 heads
when you flip the coin a fixed number of times. The partition that was useful
there will be useful here.

6. Let X be a discrete RV with the geometric distribution.
(a) Compute P(X > n).
(b) Show that P(X > n + k|X > n) = P(X > k). Because of this result it
is often said that the geometric distribution is “memoryless.”
(c) (Exposition) The geometric distribution may be realized by flipping a
biased coin until we get heads. (X is the total number of flips needed.)
Explain why the equation in part (b) is “obvious” in this experiment.

7. The experiment is that we roll a six-sided die ten times. We let X be the
sum of the ten rolls. We want to compute E[X|X > 30]. In principle we
can compute this analytically, but it will be a real mess. So the point of this
problem is to estimate it by simulation, i.e., write an R program to estimate
it. To get you started, the R program on the web does the experiment of
simulating a sequence of ten rolls and computing X. It does the experiment
20 times and prints out both the sequences of 10 rolls and their sums.

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