Publications by Tora Mullings
605 - Discussion week 11
The quakes data set contains information about the locations of earthquakes near Fiji. head(quakes) ## lat long depth mag stations ## 1 -20.42 181.62 562 4.8 41 ## 2 -20.62 181.03 650 4.2 15 ## 3 -26.00 184.10 42 5.4 43 ## 4 -17.97 181.66 626 4.1 19 ## 5 -20.42 181.96 649 4.0 11 ## 6 -19.68 184...
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DATA 605 - Discussion Week 10
Problem 2 in Section 11.2 Show that Example 11.7 is an absorbing Markov chain. An absorbing Markov chain is a Markov chain in which it is impossible to leave some states once entered. However, this is only one of the prerequisites for a Markov chain to be an absorbing Markov chain. In order for it to be an absorbing Markov chain, all other tran...
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HW 10 - 605
Smith is in jail and has 1 dollar; he can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him. If Smith bets A dollars, he wins A dollars with probability .4 and loses A dollars with probability .6. Find the probability that he wins 8 dollars before losing all of his money if he bets 1 dollar each time (timid ...
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HW 9 - 605
1. Problem 11 in 9.3 Find \(P(Y_{365} >= 100)\) \[ S_n = X_1 + X_2 + ... + X_n \] \[ Y_{365} = Y_1 + X_1 + X_2 + ... + X_n \] We are given that \[ Y_1 = 100 \] So we can substitute its value: \[ Y_{365} = 100 + S_{364} \] Now we can find \(P(100 + S_{364} \ge 100)\) \[ P(S_{364} \ge 0) \] From this, we know that \(n = 364\). And we are g...
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DATA 605 - Discussion Week 9
This was the 2nd problem from the HW, to use the moment generating function to derive the expected value and variance for the binomial distribution. Moment generating function (MGF) for a random variable X: \[ M_X(t) = E[e^{tX}] \] For a binomial distribution, the probability mass function (PMF) is \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] ...
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605 - HW 8
p. 303 11 A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? There are a 100 lightbulbs and the first one that will burn out is the minimum of the exponential random variables. Sum the lambdas (rate parameters) to get this minimum’s lambd...
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605 - Discussion week 8
A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? There are a 100 lightbulbs and the first one that will burn out is the minimum of the exponential random variables. Sum the lambdas (rate parameters) to get this minimum’s lambda. All thes...
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605 - HW 7
1 Let \(X_1, X_2, \ldots, X_n\) be \(n\) mutually independent random variables, each of which is uniformly distributed on the integers from 1 to \(k\). Let \(Y\) denote the minimum of the \(X_i\)’s. We want to find the distribution of \(Y\). \[ P(Y = y) = \left(\frac{y}{k}\right)^n - \left(\frac{y-1}{k}\right)^n \] for \(y = 1, 2, \ldots, k\...
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DATA 605 - Discussion Week 7
Solution to Question 1 from HW 7 Let \(X_1, X_2, \ldots, X_n\) be \(n\) mutually independent random variables, each of which is uniformly distributed on the integers from 1 to \(k\). Let \(Y\) denote the minimum of the \(X_i\)’s. We want to find the distribution of \(Y\). \[ P(Y = y) = \left(\frac{y}{k}\right)^n - \left(\frac{y-1}{k}\right)^...
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605 - HW6
1 A bag contains 5 green and 7 red jellybeans. How many ways can 5 jellybeans be withdrawn from the bag so that the number of green ones withdrawn will be less than 2? A breakdown of the different cases: 0 green jellybeans and 5 red jellybeans. 1 green jellybean and 4 red jellybeans. Number of ways to choose 0 green jellybeans from 5: \(\binom{...
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