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Midtermexam 2

The document outlines the instructions and guidelines for Midterm Exam #2 for STAT/MA 41600, including permitted calculators, grading criteria, and requirements for showing work. It contains five questions related to probability density functions and random variables, with specific tasks for each question. Additionally, it emphasizes the importance of justifying solutions and provides formulas for Gamma random variables, although they are not needed for this exam.

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Willy Kheed
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39 views7 pages

Midtermexam 2

The document outlines the instructions and guidelines for Midterm Exam #2 for STAT/MA 41600, including permitted calculators, grading criteria, and requirements for showing work. It contains five questions related to probability density functions and random variables, with specific tasks for each question. Additionally, it emphasizes the importance of justifying solutions and provides formulas for Gamma random variables, although they are not needed for this exam.

Uploaded by

Willy Kheed
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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STAT/MA 41600

Midterm Exam #2: November 21, 2014

Name

Purdue student ID (10 digits)

1. The testing booklet contains 5 questions, which are all weighted evenly (i.e., each question
is worth 1/5 of the midterm exam grade).

2. Permitted Texas Instruments calculators:


BA-35
BA II Plus*
BA II Plus Professional Edition*
TI-30XS MultiView*
TI-30Xa
TI-30XIIS*
TI-30XIIB*
TI-30XB MultiView*
*The memory of the calculator should be cleared at the start of the exam.

3. Circle your final answer in your booklet; otherwise, no credit may be given.

4. There is no penalty for guessing or partial work.

5. Show all your work in the exam booklet. If the majority of questions are answered cor-
rectly, but insufficient work is given, the exam could be considered for academic misconduct.
Therefore, you should show all your work and justify your solutions in the exam booklet.

6. Extra sheets of paper are available from the proctor.

You will not need these formulas on this particular midterm exam, but nonetheless,
Dr Ward promised to mention that a Gamma random variable with parameters λ and r has
probability density function:
( r
λ
(r−1)!
xr−1 e−λx , for x > 0,
fX (x) =
0 otherwise,
and the cumulative distribution function (CDF) is:

r−1
1 − e−λx P (λx)j , for x > 0,

j!
FX (x) = j=0

0 otherwise.
1. Suppose X and Y have joint probability density function

fX,Y (x, y) = 12e−3x−4y

for x > 0 and y > 0; and fX,Y (x, y) = 0 otherwise. Compute P (Y > X).

1
2. Suppose X and Y have joint probability density function

fX,Y (x, y) = 60e−4x−6y

for 0 < x < y; and fX,Y (x, y) = 0 otherwise. (Note that X and Y are not independent, since
we are insisting that X < Y . Since X < Y , then X and Y are defined in the portion of the
first quadrant that is above the line y = x.)
Find E(X).

2
3. Suppose U and V are independent, continuous random variables, each uniformly dis-
tributed on the interval [0, 10]. Define X = max(U, V ).
3a. Find the cumulative distribution function of X.

3b. Find the probability density function of X.

3c. Check your own work: Give a brief justification for why your answer to 3b is a valid
probability density function.

3
4. Suppose that U1 , . . . , U200 are independent, continuous random variables, each of which
is Uniformly distributed on the interval [0, 6].
4a. Compute E(Uj ) for a specific, fixed j in the range 1 ≤ j ≤ 200.

4b. Compute E(Uj2 ) for a specific, fixed j in the range 1 ≤ j ≤ 200.

4c. Use your answers to 4a and 4b to compute Var (Uj ) for a specific, fixed j in the range
1 ≤ j ≤ 200.

4d. Find a good approximation for P (U1 + · · · + U200 < 625).

4
5. If X and Y are two independent Poisson random variables that each have parameters
λ = 450, compute a good estimate for P (Y − X ≥ 20).

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