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Exam 3 for Math 361, Fall 2007: Probability Distributions and Moments - Prof. Richard B. S, Exams of Probability and Statistics

University of Illinois - Urbana-ChampaignProbability and Statistics
Prof. Richard B. Sowers

The solutions to exam 3 for math 361, section d13 and d14, held on november 12, 2007. The exam covers topics such as continuous uniform distributions, normal distributions, and moments. Students are expected to compute probabilities, cumulative distribution functions, densities, and expected values.

Typology: Exams

2010/2011

Uploaded on 06/28/2011

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Math 361, Section D13 and D14, Fall 2007
Exam 3, November 12
1. 50 points Suppose that Xis a continuous uniform(0,1) random variable; i.e., it has
density
fX(t) = (1 if 0 < t < 1
0 else
Define Ydef
=X3.
(a) 10 points Compute PY≤1
5.
(b) 15 points Compute the cumulative distribution FYof Y
(c) 15 points Compute the density fYof Y, if it exists. If the density does not exist,
explain why.
(d) 10 points Compute E[Y].
2. 20 points This is essentially question 21 on page 249 of the book. We will model the
weight (in pounds) of 25-year old men as normal with mean µ= 170 and variance
σ2= 25. You should use the attached table of values of Gaussian integrals. Go as far as
possible without doing messy math by hand.
(a) 10 points What percentage of 25-year old men weigh more than 165 pounds?
(b) 10 points What percentage of 25-year old men over 165 pounds are less than 180
pounds?
3. 30 points This is essentially question 7 on page 248 of the book. The density of a
continuous random variable Xis given by
fX(t) = (a+bx3if 0 < x < 1
0 else
If E[X] = 3
5, compute aand b.
R. Sowers 1
pf3

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Download Exam 3 for Math 361, Fall 2007: Probability Distributions and Moments - Prof. Richard B. S and more Exams Probability and Statistics in PDF only on Docsity!

Math 361, Section D13 and D14, Fall 2007

Exam 3, November 12

  1. 50 points Suppose that X is a continuous uniform(0, 1) random variable; i.e., it has density

fX (t) =

1 if 0 < t < 1

0 else

Define Y

def = X 3 .

(a) 10 points Compute P

Y ≤

1 5

(b) 15 points Compute the cumulative distribution FY of Y

(c) 15 points Compute the density fY of Y , if it exists. If the density does not exist, explain why.

(d) 10 points Compute E[Y ].

  1. 20 points This is essentially question 21 on page 249 of the book. We will model the weight (in pounds) of 25-year old men as normal with mean μ = 170 and variance σ^2 = 25. You should use the attached table of values of Gaussian integrals. Go as far as possible without doing messy math by hand.

(a) 10 points What percentage of 25-year old men weigh more than 165 pounds?

(b) 10 points What percentage of 25-year old men over 165 pounds are less than 180 pounds?

  1. 30 points This is essentially question 7 on page 248 of the book. The density of a continuous random variable X is given by

fX (t) =

a + bx^3 if 0 < x < 1

0 else

If E[X] = 3 5 , compute^ a^ and^ b.

Answers

  1. (a)

P

Y ≤

1 5

= P

X

3 ≤ 1 5

= P

X ≤

1 5

t=−∞

fY (t)dt = (1/5) 1 / 3 .

(b)

FY (t) = P {Y ≤ t} = P

X

3 ≤ t

= P

X ≤ t 1 / 3

∫ (^) t 1 / 3

s=−∞

fY (s)ds

0 if t 1 / 3 < 0

t^1 /^3 if 0 ≤ t^1 /^3 < 1

1 if t^1 /^3 ≥ 1

0 if t < 0

t^1 /^3 if 0 ≤ t < 1

1 if t ≥ 1

(c)

fY (t) = F ′ Y (t) =

1 3 t

− 2 / 3 if 0 < t < 1

0 else

(d)

E[Y ] = E[X

3 ] =

t=−∞

t

3 fX (t)dt =

t=

t

3 dt =

or equivalently

E[Y ] =

t=−∞

tfY (t)dt =

t=

t

t − 2 / 3

dt =

t=

t 1 / 3 dt =

  1. Let ξ be normal with mean 0 and variance 1.

(a)

P{X ≥ 165 } = P{ 5 ξ + 170 ≥ 165 } = P{ξ ≥ − 1 } = P{−ξ ≥ − 1 }

= P{ξ ≤ 1 } = 0. 8413

(b)

P{X ≤ 180 |X ≥ 165 } =

P{ 165 ≤ X ≤ 180 }

P{X ≥ 165 }

We calculate that

P{ 165 ≤ X ≤ 180 } = P{X ≤ 180 } − P{X < 165 }

We then compute that

P{X ≤ 180 } = P{ 5 ξ + 170 ≤ 180 } = P{ξ ≤ 2 } = 0. 9772.