Problem 2. Suppose a particular kind of atom has a half-life of 1 year. This means that these atomsdecay (transform) into another type of atom (or isotope). The lifetime of each single atom is assumedto have an exponential distribution with some rate > 0, which is determined by the half-life. Thehalf-life is the time h by which the atom has a 1/2 probability of decaying. That is, if T is the survivaltime of an atom, P(T> h) = 1/2. Atoms in a large 'population' are assumed to decay independently.(1) Find by using that the half-life is 1 year.(2) Find the probability that an atom of this type survives at least 5 years.(3) Find the time at which the expected number of atoms is 10% of the original. Hint: use theindictor method with indicators of the form I,,t of the event that atom j has survived at leasttime t.(4) Find the probability that in fact none of the 1024 original atoms remains after the timecalculated in (3).

Explained in detail in the below.Explanation:1) Find A by using that the half-life is 1 year. The…

Answered: Problem 2. Suppose a particular kind of…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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The following table gives the millions of metric tons of carbon dioxide emissions in a certain country for selected years from 2010 and projected to 2032. 2012 2014 2016 2018 2020 CO2 Emissions 338.5 364.5 397.1 423.8 452.1 498.4 Year 2010 Year 2022 2024 2026 2028 2030 CO2 Emissions 557.2 592.9 628.7 662.1 707.1 (a) Create a linear function that models these data, with x as the number of years past 2010 and y as the millions of metric tons of carbon dioxide emissions. (Round all numerical values to two decimal places.) y(x) = 2032 741.7 (b) Find the model's estimate for the 2024 data point. (Round your answer to two decimal places.) million metric tons Interpret the slope of the linear model. For each year since ---Select--- (c) Find the slope of the linear model. (Round your answer to two decimal places.) , carbon dioxide emissions in the U.S. are expected to change by million metric tons.
Researchers have created every possible "knockout" line in yeast. Each line has exactly one gene deleted and all the other genes present (Steinmetz et al. 2002). The growth rate—how fast the number of cells increases per hour—of each of these yeast lines has also been measured, expressed as a multiple of the growth rate of the wild type that has all the genes present. In other words, a growth rate greater than 11 means that a given knockout line grows faster than the wild type, whereas a growth rate less than 11 means it grows more slowly. The growth rate of a random sample of knockout lines is 0.86, 1.02, 1.02, 1.01, 1.02, 1, 0.99, 1.01, 0.91, 0.83, 1.01 What is the standard deviation and variance of growth rate for this sample?
6. A home improvement store carries two models of power drills. Lifetime (i.e., time to failure) of Model 1 drills are exponentially distributed with a rate of of 0.15 failures per year, whereas Model 2 drills have a lifetime that is log-normally distributed with parameters u = 0.75 and o = 1.2. (a) Which model of power drill is more likely to fail within a year? (b) Which brand produces cables that are more likely to last at least 2 years? (c) Which model of power drill has longer expected lifetime?
Question no. 10 A machine is made up of two components that operate independently. The lifetime T; (in days) of component i has an exponential distribution with parameter Ai, for i = 1,2. Suppose that the two components are placed in parallel and that A1 A2 = In 2. When the machine breaks down, the two components are replaced by new ones at the beginning of the following day. Let Xn be the number of components that operate at the end of n days. Then the stochastic pro- cess {Xn,n probability matrix. 0,1,...} is a Markov chain. Calculate its one-step transition
Question 10: a) Suppose you have a log utility function. Suppose your current wealth W = $300,000. How much would you pay today (A) to insure yourself against a 10% probability of a $150,000 loss? b) Suppose you have a log utility function. Suppose your current wealth W = $300,000. How much would you pay today (A) to avoid a "fair gamble" with a 50% and 50% payoff of +$1000 or -$1000?
6. A small company can employ 3 workers. Each worker independently stays on the job for an exponentially distributed time with a mean of one year and then quits. When a worker quits, it takes the company an exponentially distributed time with a mean of 1/10 of a year to hire a replacement for that worker (independently of how long it takes to replace other workers). If the company takes in $1000 per day in revenue when it has 3 workers, $800 per day when it has 2 workers, and $500 per day when it has 0 or 1 workers, what is the company's long-run average daily revenue?
Question 25 The population of deer in a certain area of Stanly county grows proportional to itself. The populat of deer in 2000 was found to be 145, and by 2010 the population had frown to 185. i. Find the growth constant k (rounded to 6 decimal places). k= ii. Find the expected number of deer in 2013 (rounded down to the nearest integer) Number of deer = iii. Find the rate of growth in 2015 (rounded to 3 decimal places) Rate of growth = deer per year

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