https://dananne.org/dw/ 2026-05-06T07:03:32+00:00 https://dananne.org/dw/ https://dananne.org/dw/lib/tpl/codowik/images/favicon.ico text/html 2015-08-31T05:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-1&rev=1441024872&do=diff Mathematics 340 - Fall 2015 Homework * Suppose a coin is tossed three times. * Find a sample space for this experiment. * Find the event E that at least one toss is a head. * Suppose a fuse is tested until it fails. * Find a sample space for this experiment.$E_k$$k$$k = 1, 2, \ldots$$E_1 \cup E_2 \cup E_3$$E_1 \cap E_2 \cap E_3$$E_3 \cap E_2^c$$\cup_{k=1}^\infty E_k$$\cap_{k=1}^\infty E_k$$\Omega = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$$E = \{HHH, HHT, HTH, THH, HTT, THT, TT… text/html 2015-08-31T05:44:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-2&rev=1441025041&do=diff Mathematics 340 - Fall 2015 Homework Exercises 1.9: 1, 2, 3, 5, 11, 21, 22, 23, 24(b), 26, 31(a)(b) For Problem Set due 9 September: 11 text/html 2015-08-31T05:43:41+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-3&rev=1441025021&do=diff Mathematics 340 - Fall 2015 Homework Exercises 1.9: 4, 8, 9, 10, 12, 14, 29, 30, 32, 33 For Problem Set due 9 September: 10, 33 text/html 2015-09-01T10:47:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-4&rev=1441129674&do=diff Mathematics 340 - Fall 2015 Homework Exercises 1.9: 17, 18(a) For Problem Set due 9 September: 17 text/html 2015-09-03T11:46:04+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-5&rev=1441305964&do=diff Mathematics 340 - Fall 2015 Homework Exercises 1.9: 42, 43, 47, 48, 50 For Problem Set due 16 September: 42, 47 text/html 2015-09-08T09:32:17+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-6&rev=1441729937&do=diff Mathematics 340 - Fall 2015 Homework Exercises 2.11: 5, 15, 16, 18, 20 For Problem Set due 16 September: 20 text/html 2015-09-10T11:51:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-7&rev=1441911115&do=diff Mathematics 340 - Fall 2015 Homework Exercises 2.11: 1, 2, 3, 4, 8, 11, 21, 57(a)(b), 60 For Problem Set due 16 September: 8, 11 text/html 2015-09-13T11:34:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-8&rev=1442169249&do=diff Mathematics 340 - Fall 2015 Homework Exercises 2.11: 6, 30, 31, 35, 36, 55, 57(c) text/html 2015-09-20T09:34:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-9&rev=1442766841&do=diff Mathematics 340 - Fall 2015 Homework Exercises 3.12: 1, 5(a), 6, 8, 10 Also: For each of the following, find the probability mass function for the given random variable: * Suppose 4 chips are drawn, without replacement, from an urn with 3 red chips and 2 blue chips. Let $X$$X$$Y$$p_X(x) = \begin{cases}\frac{3}{5},& \text{if } x = 2, \\ \frac{2}{5},& \text{if } x = 3, \\ 0,& \text{otherwise}.\end{cases}$$p_X(x) = \begin{cases}\frac{1}{16},& \text{if } x = 0, \\ \frac{1}{4},& \text{if } x =… text/html 2015-09-17T06:29:25+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-10&rev=1442496565&do=diff Mathematics 340 - Fall 2015 Homework Exercises 3.12: 2, 17, 18, 19, 20, 21, 22, 23, 28 text/html 2015-10-08T17:41:19+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-11&rev=1444351279&do=diff Mathematics 340 - Fall 2015 Homework Exercises 3.12: 4, 15, 30(a)(b), 31(a), 32 Also: For each of the following, find the cumulative distribution function for the given random variable: * Suppose 4 chips are drawn, without replacement, from an urn with 3 red chips and 2 blue chips. Let $X$$X$$Y$$F_X(x) = \begin{cases}0,& \text{if } x < 2, \\ \frac{3}{5},& \text{if } 2 \le x < 3, \\ 1,& \text{if } x \ge 3, \\ 0.\end{cases}$$F_X(x) = \begin{cases}0,& \text{if } x < 0, \\ \frac{1}{16},& \tex… text/html 2015-10-11T09:03:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-12&rev=1444579390&do=diff Mathematics 340 - Fall 2015 Homework Exercises 3.12: 3, 41 Also: * Suppose $X$ has a uniform distribution on the integers $-3, -2, -1, 0, 1, 2, 3$. Find the probability mass function for $Z = X^2$. * Suppose $X$ and $Y$ are independent random variables, both with uniform distributions on the integers $-2, -1, 0, 1, 2$$T = X + Y$$X$$Y$$-2, -1, 0, 1, 2$$S = X - Y$$X$$-2, -1, 0, 1, 2$$Z = X^2 - X$$X$$Y$$1, 2, \ldots, 10$$W = \max(X, Y)$$U = \min(X, Y)$$W$$U$$P(U < W)$$P(U \le W)$$U$$W$$p_Z… text/html 2015-09-29T10:27:33+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-13&rev=1443547653&do=diff Mathematics 340 - Fall 2015 Homework Exercises 3.12: 14, 16, 24, 29, 33, 35 For Problem Set due 5 October: 24 text/html 2015-10-01T07:51:37+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-14&rev=1443711097&do=diff Mathematics 340 - Fall 2015 Homework Exercises 4.12: 1, 2, 3, 11, 12, 14 For problems 1, 2, and 14 compute only the mean For Problem Set due 14 October: 14 text/html 2015-10-04T12:00:18+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-15&rev=1443985218&do=diff Mathematics 340 - Fall 2015 Homework Exercises 4.12: 17, 18, 24, 25, 29, 58, 59 For Problem Set due 14 October: 25, 58 text/html 2015-10-06T08:51:37+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-16&rev=1444146697&do=diff Mathematics 340 - Fall 2015 Homework Exercises 4.12: 1, 2, 6, 7, 14 For Problem Set due 14 October: 6, 14 text/html 2015-10-20T05:36:40+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-17&rev=1445344600&do=diff Mathematics 340 - Fall 2015 Homework Exercises 4.12: 21, 28, 60, 65, 76(a)(b) Also: Suppose 1 out of every 10,000 people have a certain disease. Let $X$ be the number of people in a sample of 20,000 that have disease. * What is $\text{E}(X)$? * Find, exactly, $P(X = 0)$$P(X = 1)$$P(X = 2)$$P(X > 3)$$E(X) = 200000 \cdot \dfrac{1}{10000} = 2$$P(X = 0) = (0.9999)^{20000} = 0.135322$$P(X = 1) = 20000 \cdot (0.0001) \cdot (0.9999)^{19999} = 0.270671$$P(X = 2) = \binom{20000}{2} \cdot (0.0001… text/html 2015-10-15T06:00:52+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-18&rev=1444914052&do=diff Mathematics 340 - Fall 2015 Homework Exercises 5.10: 1, 5, 7(a), 8, 9, 11 text/html 2015-11-16T06:06:25+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-19&rev=1447682785&do=diff Mathematics 340 - Fall 2015 Homework Exercises 5.10: 21, 22, 24, 25, 26 Note: use the table to find the normal probabilities in the exercises. Also: * Suppose $X \sim \mathcal{N}(10, 25)$. Find $P(0 < X < 10)$, $P(X \ge 9)$, and $P(8 \le X < 14)$. * Suppose $Z \sim \mathcal{N}(0, 1)$ and $W = Z^2$. Show that the probability density function of $W$ is $g_W(w) = \begin{cases}\frac{1}{\sqrt{2\pi w}}e^{-\frac{w}{2}},& \text{if } w > 0, \\ 0,& \text{otherwise}\end{cases}$$Z$$W$$g_W$$E(Z^2)$$… text/html 2015-11-15T03:32:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-20&rev=1447587162&do=diff Mathematics 340 - Fall 2015 Homework Exercises 5.10: 30, 40(a)(b), 44, 46, 47, 48, 49, 50, 51(a), 54 Hint: 49 will help with 48(b). Note: use the table to find the normal probabilities in the exercises. Also: * Suppose the lifetime of a certain tire has a normal distribution with mean 50,000 miles and standard deviation 5000 miles. Find:$a$$a$$a$$a$ text/html 2015-11-03T18:14:46+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-21&rev=1446603286&do=diff Mathematics 340 - Fall 2015 Homework Exercises 6.10: 18, 21 Also: * Suppose $X$ is uniform on $(a, b)$. Show that the moment generating function of $X$ is $M_X(t) = \frac{e^{tb} - e^{ta}}{t(b - a)}$. * Suppose $X$ is geometric with probability of success $p$ and let $q = 1 - p$. Show that the moment generating function of $X$$M_X(t) = \frac{p}{1 - qe^t}$$qe^t < 1$$X$$M_X(t) = \frac{1}{(1 - 2t)^3}$$t < \frac{1}{2}$$X$$X \sim \text{Expo}(\lambda)$$X$$k$$X$$\mu_k = \frac{k!}{\lambda^k}$$M_X… text/html 2015-10-30T05:46:17+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-22&rev=1446209177&do=diff Mathematics 340 - Fall 2015 Homework Exercises 6.10: 13, 14, 15, 19, 20, 23, 24 For Problem Set due 11 November: 15, 23 text/html 2015-11-01T10:32:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-23&rev=1446402721&do=diff Mathematics 340 - Fall 2015 Homework Exercises 7.8: 1, 4(a)(b), 8, 16(a)(c), 17(a)(c), 21 For Problem Set due 11 November: 17(a)(c) text/html 2015-11-03T07:09:37+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-24&rev=1446563377&do=diff Mathematics 340 - Fall 2015 Homework Exercises 7.8: 4(c)(d), 5(a), 16(b)(d), 17(b)(d), 18, 22, 30 For Problem Set due 11 November: 17 text/html 2015-11-12T12:16:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-25&rev=1447359361&do=diff Mathematics 340 - Fall 2015 Homework Exercises 7.8: 24, 31, 32(a), 35, 36 Also: * Suppose $X$ and $Y$ are jointly continuous random variables with joint probability density function $f_{X,Y}(x,y) = \begin{cases}8xy,& \text{if } 0 < x < y < 1,\\ 0,& \text{otherwise}.\end{cases}$ Find each of the following: * $E(X)$ * $E(Y^2)$ * $E(XY^3)$ * Suppose $X$ and $Y$ are independent standard normal random variables. Find the distribution of $W = \sqrt{X^2 + Y^2}$$X$$Y$$Y$$(0, 1)$$X$$(0,… text/html 2015-11-10T16:40:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-26&rev=1447202457&do=diff Mathematics 340 - Fall 2015 Homework Exercises 7.8: 34 Also: Find $\text{Corr}(X, Y)$ for each of the following joint probability density functions: * $f_{X,Y}(x, y) = \begin{cases} 8xy,& \text{if } 0 < y < x < 1,\\ 0,& \text{otherwise}.\end{cases}$. * $f_{X,Y}(x, y) = \begin{cases} x + y,& \text{if } 0 < y < 1, 0 < x < 1,\\ 0,& \text{otherwise}.\end{cases}$. * $f_{X,Y}(x, y) = \begin{cases} 24xy,& \text{if } x > 0, y > 0, x + y < 1,\\ 0,& \text{otherwise}.\end{cases}$. * $f_{X,Y}(x… text/html 2015-11-11T05:51:25+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-27&rev=1447249885&do=diff Mathematics 340 - Fall 2015 Homework Exercises 8.9: 30, 37, 52 Also: * Let $Z$ be a standard normal random variable. Show that $Z^2$ has a gamma distribution. What are the parameters? Hint: We did this in a previous exercise. * Let $Z_1$ and $Z_2$ be independent identically distributed standard normal random variables. Show that $Z_1^2 + Z_2^2$ text/html 2015-11-15T06:57:51+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-28&rev=1447599471&do=diff Mathematics 340 - Fall 2015 Homework Exercises 10.7: 1, 2 Hints: For Exercise 1, note that $f(p) = p(1 - p)$ is maximized on $[0, 1]$ when $p = \frac{1}{2}$. For Exercise 2, the sample mean of $X_1, X_2, \ldots, X_n$ is $\bar{X} = \dfrac{X_1 + X_2 + \cdots + X_n}{n}$. Also: * Let $X$ be binomial with parameters $n = 4$ and $p = \frac{1}{2}$. Use Chebyshev's inequality to find an upper bound $P(|X - 2| \ge 2)$$Z$$P(|Z| \ge 4)$$P(|X - 2| \ge 2) \le \frac{1}{4}$$P(|X - 2| \ge 2) = \frac{1}{… text/html 2015-12-15T09:17:48+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-31&rev=1450199868&do=diff Mathematics 340 - Fall 2015 Homework Exercises 10.7: 18, 21 Also: * A fair coin is tossed 1000 times. Use the Central Limit Theorem to approximate the probability that the number of heads is strictly between 480 and 509. * A basketball player makes 80% of her free throws. Use the Central Limit Theorem to approximate the probability that in 80 attempts she makes at least 60. text/html 2015-12-05T07:12:20+00:00 Anonymous (anonymous@undisclosed.example.com) https://dananne.org/dw/doku.php?id=math-340:m340-f15-hw:hw-32&rev=1449328340&do=diff Mathematics 340 - Fall 2015 Homework * Suppose $X$ has a chi-square distribution with $200$ degrees of freedom. Use the Central Limit theorem to estimate the probability that $X$ is greater than $215$. Answer: * 0.2266