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Probability and Statistics Exam Review, Exams of Data Analysis & Statistical Methods

University of Illinois - ChicagoData Analysis & Statistical Methods

Review problems and solutions for topics including joint and conditional probability, expectation and variance, binomial and poisson distributions, and normal and weibull distributions.

Typology: Exams

2011/2012

Uploaded on 05/18/2012

koofers-user-380
koofers-user-380 🇺🇸

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STAT 381 Answers to Review for Exam 2
1. Suppose we draw 2 balls out of an urn with 8 red, 6 blue, and 4 green balls. Let
Xbe the number of red balls we get and Ythe number of blue balls.
(a) Find the joint pmf of Xand Y.
Y X = 0 1 2 fY
0 6/153 32/153 28/153 66/153
1 24/153 48/153 0 72/153
2 15/153 0 0 15/153
fX45/153 80/153 28/153
(b) Find the marginals of Xand Y.
(c) Find the conditional pmf of fY|x(y|x= 1)
Y|X= 1 0 1 2
P(Y|X= 1) 32/80 48/80 0
(d) Find P(Y= 1|X= 1) = 48/80
2. Suppose P(X=x, Y =y) = c(x+y) for x, y = 0,1,2,3.
(a) What value of cwill make this a joint density? c= 1/48
(b) What is P(X > Y )? 3/8
(c) Find E(X), V (X), cov (X, Y )
Marginal fX(x) is X=k0123
P(X=k) 3/24 5/24 7/24 9/24
E(X) = 23/12, V (X) = 155/144, E(XY ) = 168/48 = 7/2, C ov(X , Y ) =
−25/144.
3. Suppose Xand Yhas joint density f(x, y) = c(x+y) for 0 < x, y < 1.
(a) What is c?c= 1
(b) What is P(X < 1/2)? 3/8
(c) What is P(X+Y > 1/2)? 23/24
(d) Find E(X), V (X), cov (X, Y )
E(X) = R1
0R1
0x(x+y)dxdy = 7/12, V (X) = R1
0R1
0x2(x+y)dxdy −(7/12)2=
11/144
Cov(X, Y ) = E(X Y )−(7/12)2=R1
0R1
0xy(x+y)dxdy −(7/12)2=−1/144
4. A chromosome mutation believed to be linked with color blindness is known to
occur, on the average, once in every 10,000 births. If 20,000 babies are born this
year in a certain city, what is the probability that at least one will develop color
blindness? What is the exact probability model that applies here?
Binomial with n= 20000, p = 10−4. P (X≥1) = 0.8647
pf3
pf4

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STAT 381 Answers to Review for Exam 2

  1. Suppose we draw 2 balls out of an urn with 8 red, 6 blue, and 4 green balls. Let X be the number of red balls we get and Y the number of blue balls. (a) Find the joint pmf of X and Y. Y X = 0 1 2 fY 0 6 / 153 32 / 153 28 / 153 66 / 153 1 24 / 153 48 / 153 0 72 / 153 2 15 / 153 0 0 15 / 153 fX 45 / 153 80 / 153 28 / 153 (b) Find the marginals of X and Y. (c) Find the conditional pmf of fY |x(y|x = 1) Y |X = 1 0 1 2 P (Y |X = 1) 32 / 80 48 / 80 0 (d) Find P (Y = 1|X = 1) = 48/ 80
  2. Suppose P (X = x, Y = y) = c(x + y) for x, y = 0, 1 , 2 , 3. (a) What value of c will make this a joint density? c = 1/ 48 (b) What is P (X > Y )? 3/ 8 (c) Find E(X), V (X), cov(X, Y ) Marginal fX (x) is X P (X=^ k=^ k) 3 / 240 5 / 241 7 / 242 9 / 243 E(X) = 23 / 12 , V (X) = 155 / 144 , E(XY ) = 168 / 48 = 7 / 2 , Cov(X, Y ) = − 25 / 144.
  3. Suppose X and Y has joint density f (x, y) = c(x + y) for 0 < x, y < 1. (a) What is c? c = 1 (b) What is P (X < 1 /2)? 3/ 8 (c) What is P (X + Y > 1 /2)? 23/ 24 (d) Find E(X), V (X), cov(X, Y ) E(X) = ∫^01 ∫^01 x(x + y)dxdy = 7/ 12 , V (X) = ∫^01 ∫^01 x^2 (x + y)dxdy − (7/12)^2 = 11 / 144 Cov(X, Y ) = E(XY ) − (7/12)^2 = ∫^01 ∫^01 xy(x + y)dxdy − (7/12)^2 = − 1 / 144
  4. A chromosome mutation believed to be linked with color blindness is known to occur, on the average, once in every 10,000 births. If 20,000 babies are born this year in a certain city, what is the probability that at least one will develop color blindness? What is the exact probability model that applies here? Binomial with n = 20000, p = 10−^4. P (X ≥ 1) = 0. 8647
  1. In a certain published book of 520 pages, 390 typographical errors occur. What is the probability that one page, selected randomly by printer as a sample of her work, will be free from errors? Poisson with λ = 390/520 errors per page. P (X = 0) = e−λ^ = 0. 4724
  2. The p.d.f of a random variable X is given by f (x) = ex, for x < 0 (a) Find E(X) = ∫^ −∞∞ xf (x)dx = − 1 (b) Find E(e^3 X/^2 ) = ∫^ −∞∞ e^3 x/^2 f (x)dx = 2/ 5
  3. The p.d.f of a random variable X is given by f (x) = kxe−x^2 , for x > 0 (a) Find the constant k. k = 2 (b) Find the distribution function F (x). F (x) =

{ 1 − e−x^2 for x > 0 0 for x ≤ 0 (Note: Don’t forget the domain.) (c) Find P (X > 4) = e−^16

  1. The p.d.f of a random variable X is given by f (x) = √cx, for 0 < x < 4

(a) Find the constant c. c = 1/ 4 (b) Find the distribution function F (x).

F (x) = 0 , x < 0 = 12 √x, 0 ≤ x < 4 = 1 , x ≥ 4 (c) Find P (X < 14 ) and P (X > 1). P (X < 14 ) = 1/4, P (X > 1) = 1 − F (1) = 1/ 2 (d) Find the mean E(X) and the variance V (X). E(X) = 4/ 3 , V (X) = 64/ 45

  1. The distribution function of a r.v X is given by

F (x) =

{ 1 − (1 + x)e−x^ for x ≥ 0 0 for x < 0 Find (a) P (X < 2) = 1 − 3 e−^2 (b) P (1 < X < 3) = 2e−^1 − 4 e−^3 (c) P (X > 4) = 5e−^4 (d) the p.d.f of X: f (x) = xe−x, x > 0

(d) P (X < 20) = 0.015%

  1. If a r.v. X satisfies a Normal distribution with mean 2, variance 9 and a r.v. Y satisfies a normal distribution with mean 3, variance 4. Suppose X and Y are independent. (a) What is the mean and variance of X + Y? E(X + Y ) = 5, V (X + Y ) = 13 (b) What is the mean and variance of X − Y? E(X − Y ) = − 1 , V (X − Y ) = V (X) + (−1)^2 V (Y ) = 13 (c) What is the mean and variance of 2X − 3 Y? E(2X − 3 Y ) = − 5 , V (2X − 3 Y ) = 22 V (X) + (−3)^2 V (Y ) = 72
  2. If a r.v.X satisfies a Poisson distribution with mean 2 and a r.v. Y satisfies a Poison distribution with mean 3. Suppose X and Y are independent. (a) What are the variances of X and Y? V (X) = 2, V (Y ) = 3 (b) What is the mean and variance of X + Y? E(X + Y ) = 5, V (X + Y ) = 5