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Answer all questions as the instruction in pdf. No use of chat gpt.
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Homework #4
STAT351.001 Win24
Due: Sunday February 25th
You do not need to print out this assignment; you may provide your work and answers on your own
separate paper (had written is sufficient).
Upload your solutions as a single PDF file to the “Homework 4” submission folder in our Canvas course.
Your submission should be titled “(your name) STAT351 Homework 4.”
2
Include all work in a neat and well organized presentation. Grading is based on the quality,
thoroughness, and correctness of the work provided.
When providing probabilities provide them as numbers between 0 and 1 (not as percentages), and keep
at least 3 decimal places.
For Expected Values, keep at least one decimal place.
_____________________________________________________________________________________
1.
Let X be a random variable with the following Probability Mass Function:
(8pts)
x
pX(x)
1
.20
2
.25
3
.20
4
.14
5
.09
6
.07
7
.04
8
.01
For parts a.-c., use the table provided above to
a.
determine ( < 5).
b.
determine ( ≥ 4).
c.
determine (2 ≤ < 7).
d.
x
FX(x)
Consider the Cumulative Probability Function, FX(x), where ( ) = ( ≤ ).
Complete the following table:
1
2
3
4
5
6
For parts e.-g., use the table you created in part d.
Be sure to show work illustrating how you used your FX(x) results.
e.
determine ( < 5).
f.
determine ( ≥ 4).
g.
determine (2 ≤ < 7).
7
8
______________________________________________________________________________
4.
Let X be a random variable with the following Probability Mass Function:
(6pts)
x
pX(x)
1
.20
a.
b.
c.
2
.25
3
.20
4
.14
5
.09
6
.07
7
.04
8
.01
Determine the mean, .
Determine the variance, .
Determine the standard deviation, .
For parts d – f, use the above random variable, X, and let Y = 4X – 20
d.
Determine P( > 0).
e.
Determine the mean of Y, .
f.
Determine the variance of Y, .
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3.
Six radar stations, operating independently, are set to detect any object flying through
(2pts)
a certain air space. Each station has a probability of 0.85 of detecting an object in this
particular air space.
Assuming that a flying object has entered this air space, let X denote the number of radar
stations that detect it.
a.
b.
Calculate .
Calculate .
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4.
Matlab exercise.
(4pts)
Often times the probability distribution of a Random Variable of interest is unknown.
In such cases simulation can be a useful tool to estimate the probability distribution.
Suppose that X1, X2, X3 are each Binomial(n=20, p=.5) independent random variables.
And let Y = minimum(X1, X2, X3).
We are interested in the probability distribution of Y.
Randomly generate N = 1000000 values for each of X1, X2, X3.
Matlab code:
>>N=1000000;
>>n=20;
>>p=.5;
>>X1=random(‘bino’, n, p, [N,1]); % repeat these steps for X2 and X3.
To create vector Y where Y = minimum(X1, X2, X3) we can use:
>>Y=min(X1,min(X2,X3)); % Produces vector: = [ ] = [min ( 1 , 2 , 3 )]
Note that Y is a discrete RV (possible values 0,1,2,…, 20).
We estimate ( ) = ( = ) by the sample proportion of times Y = y.
Create a normalized histogram of Y. “normalized” for a discrete random variable means
to express the column height as proportions (note the proportions must naturally sum to
1; a requirement for any pmf)
Use the appropriate number of bins so that the columns align with the integer values.
You may have to try a couple until you get something you like.
Promising candidates include:
>>bins=21; % (one bin for each integer 0,1,2,…,20), but this might not space the
columns quite right if the min(Y) > 0 or the max(Y) < 20. If so, you may try:
>>bins=max(Y)-min(Y)+1;
To create the normalized histogram you can use:
>> histogram(Y,bins,‘normalization’,’probability’)
Note: To normalize a discrete RV (as in this case) use ‘probability’.
To normalize a continuous RV (as you did in Homework#3) use ‘pdf’.
In addition, compute the mean and standard deviation of Y; the commands are mean(Y)
and std(Y) respectively. Note these would be denoted and sY respectively since they
are based only on our sample results they are estimates of and respectively.
For you to hand in:
a.
normalized histogram of Y
b.
mean and standard deviation of vector Y
c.
Based on your simulation estimate P( ≤ 10).
>>Prob=(sum(Y
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