4.8 Hypergeometric Distribution
A population has N individuals and r of them have a given property of interest. Consider a sample of size n
without replacement out of the population such that all groups of size n are equally likely. Let X denote the
numbe
36-226 HOMEWORK 8
Due: Monday 4/11/16 at 9:30am in class
Remember to show all work; just writing down the answer will not receive credit.
1. Exercise 10.2 in Wackerly.
2. Exercise 10.6 in Wackerly.
3. Altered Wackerly 10.17
Briefly, were interested in the
36-226 HOMEWORK 9
Due: Friday 4/22/16 at 9:30am in class
Remember to show all work; just writing down the answer will not receive credit.
1. Back to our altered Wackerly 10.17:
Recall that we were interested in the amount of breaststroke training (in mete
36-226 HOMEWORK 7
Due: Monday 3/28/16 at 9:30am in class
Remember to show all work; just writing down the answer will not receive credit.
1. Recall that we can sometimes have more than one estimator for a parameter.
Let Yi be i.i.d. Gamma(4, )
fY (y) =
y
36-226 HOMEWORK 6
Due: WEDNESDAY 3/21/16 at 9:30am in class
Remember to show all work; just writing down the answer will not receive credit.
1. Exercise 9.82 in Wackerly, et al.
2. Exercise 9.83 in Wackerly, et al.
3. Exercise 9.85 in Wackerly, et al.
4.
36-226 HOMEWORK 6 Solutions
Due: Friday 3/20/15 at 9:30am in class
Remember to show all work; just writing down the answer will not receive credit.
1. Recall that we can sometimes have more than one MOME for a parameter.
(often happens when the same param
5 Continuous Random Variables
Continuous random variables can assume an uncountable number of values. They are commonly used to model
quantities such as time, stock returns, height, weight, etc . . . While the distribution of a discrete random variable is
4.4 Geometric Distribution
Let X1 , X2 , . . . be a Bernoulli Process with parameter p. Dene Y as the smallest index i such that Xi = 1.
That is, for each w , Y (w) = miniN cfw_i : Xi (w) = 1. Y can be interpreted as the number of trials in a
Bernoulli Pr
4.5 Exercises
Exercise 75. In a certain population, 10% of people have blood type O, 40% have blood type A, 45% have blood
type B, and 5% have blood type AB. Let Y denote the number of donors who enter a blood bank on a given day
until the rst potential d
n
V ar[X] = V ar[
ISi ]
i=1
n
=
V ar[ISi ]
(Additivity of Variance, Lemma 19)
i=1
= np(1 p)
(Properties of Bernoulli(p), Lemma 32)
Example 41 (Coding). Write a code that generates a number according to the Binomial(n,p) distribution. Consider the rbernoul
4 Bernoulli Processes (a long example. . . )
4.1 Bernoulli Distribution
We say that the distribution of a random variable X is Bernoulli if X can assume values in cfw_0, 1. In certain
contexts, 0 is interpreted as a failure and 1 as a success. We say that
Finally, recall that
V ar[X] = E[X 2 ] E[X]2
(Lemma 17)
2
2
= E[X ] E[X] + E[X] E[X]
= E[X(X 1)] + E[X] E[X]2
kn
N
kn
=
N
=
(k 1)(n 1) kn
+
N 1
N
N k N n
N
N 1
Additivity of Expected Value (Lemma 14)
kn
N
2
Example 47 (Coding). Write a code that generates
4.6 Negative Binomial Distribution
Consider a sequence of independent Bernoulli trials, all having the same probability of success, p. Recall the
geometric distribution involves performing trials until the rst success. The negative binomial is a generaliz
4.12 Review of Special Discrete Distributions
1. The Binomial Random Variable - X Binomial(n, p).
X counts the number of successes out of n independent (Bernoulli) trials, each having a probability of
success p.
2. The Geometric Random Variable - X Geom(p
4.10 Poisson Distribution
The Poisson distribution is commonly used to model the occurence of rare events and can be used as an approximation of the Binomial(n, p) distribution when n is large and p is small. Formally, if X has distribution Poisson
with p
36-226 HOMEWORK 4
Due: Friday 2/12/16 at 9:30am in class
Remember to show all work; just writing down the answer will not receive credit.
1. Weve seen that
= Y and p =
V ar[
] =
2
n
, V ar[
p] =
p(1p)
.
n
Y
n
are unbiased estimates of , p respectively w