I. Simple Probabilities
1. From the specific population of numbers 1,2,3,4,5,6,7,8,9,10
a value is randomly selected, denoted X.
Let event A=X is less than 7.
Let event B=X is greater than 3.
Find
P(A) = ___________
P(B) = ___________
II. Consider the following 2 by 2 table showing the distribution
of S of events A and B:
| |
B |
not B |
| A |
.3 |
.3 |
| not A |
.4 |
0 |
1. Show the events A and B are independent or not independent.
2. Find P(A|B) =
3. Find P(A or B) =
4. Find P(B|A)=
5. Verify Bayes' rule P(A|B)=P(B|A) P(A) / P(B)
6. If 5 values are sampled independently, then what is the
probability of five consecutive values of X less than 7?
III.
The following is a simple probability distribution:
P(X=1) = 0.1
P(X=3) = 0.4
P(X=5) = 0.4
P(X=7) = 0.1
1. Find the EX=
2. Find the variance of X=
3. The joint probability distribution of X and Y is
| |
Y=1 |
Y=3 |
Y=5 |
Y=7 |
| X=1 |
.01 |
.04 |
.04 |
.01 |
| X=3 |
.04 |
.16 |
.16 |
.04 |
| X=5 |
.04 |
.16 |
.16 |
.04 |
| X=7 |
.01 |
.04 |
.04 |
.01 |
where X is one sample from the above probability distribution
and Y is another independent sample from the same probability
distribution.
a. Derive the distribution of the sum of S=X+Y from the
above joint distribution
S
P(S)
b. Using the distribution of S, find the expectation of
the sum (show calculations).
c. Using the distribution of S, find the variance of the
sum (show calculations).
d. Without reference to the distribution of S, find the
variance of S=
e. If a sample of n=10 observations are independently selected
from the above probability distribution of X, then for the
mean x-bar=S/n:
Find the E(X-bar)
Find the variance of X-bar
What is the approximate distribution of x-bar?
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