Instructions: (ODL/ F.T students) Answer all questions Show all your working to earn full marks Write your full names and computer number on your answer sheet. TIME ALLOWED: 1 Hour 19th May, 2023. Suppose we know that the number of items produced in a factory during a week is a random variable with mean 200. (a) Find a bound on the probability that this week's production will be at least 500. (b) If the variance of a week's production is known to be 60, what can be said about the probability that this week's production will be between 100 and 300? (a) The lifetime of a special type of battery is a random variable with mean 40 hours and standard deviation 20 hours. A battery is used until it fails; at which point it is replaced by a new one. Assuming a stockpile of 25 such batteries the lifetime of which are independent, find the probability that over 1100 hours of use can be obtained. (b) State and prove the weak law of large numbers. (c) (i) Let X1, X2, ..., X10 be independent Poisson random variables with mean 1. Use the Markov inequality to get a bound on P(X1 + X2 + ... + X10 ? 15). (ii) Let X1, X2, ..., X10 be independent Uniform (0, 5) random variables. Use the central limit theorem to approximate P(X1 + X2 + ... + X10 ? 15). THE END - GOOD LUCK
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Here, the mean of the number of items produced in a week is 200. So, the probability that this week's production will be at least 500 is bounded by \( \frac{200}{500} = 0.4 \). Show more…
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Ameer S.
The time required to complete a 3 -h final exam is modeled by the following pdf: $$f(x)=\left\{\begin{array}{cc}{\frac{4}{27} x^{2}(3-x)} & {0 \leq x \leq 3} \\ {0} & {\text { otherwise }}\end{array}\right.$$ Consider simulating values from this distribution using the accept-reject method with a uniform "candidate" distribution on the interval $[0,3] .$ (a) Find the smallest majorization constant $c$ so that $f(x) / g(x) \leq c$ for all $x$ in $[0,3] .[$Hint . [Hint: What is the pdf of the uniform distribution on $[0,3] ? ]$ (b) Write a program to simulate values from this distribution. (c) On the average, how many candidate values must your program generate in order to create $10,000$ "accepted" values? (d) A professor has 20 students taking her class (lucky professor!). Assume her 20 students' completion times on the final exam can be modeled as 20 independent observations from the above pdf. The professor must stay at the final exam until all 20 students are finished (i.e., until the last student leaves). Use your program in (b) to simulate the rv $L=$ time, in hours, that the professor sits proctoring her final exam to 20 students. Use your simulation to estimate $P(L \geq 35 / 12),$ the probability she will have to stay into the last 5 min of the final exam period.
Continuous Random Variables and Probability Distributions
Simulation of Continuous Random Variables
The z-scores for X values greater than the mean are negative. True False The normal distribution can be used to approximate a binomial distribution when np is less than 5. True False The z-scores for X values greater than the mean are negative. True False The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the mean? A. 120 minutes B. 150 minutes C. 135 minutes D. 270 minutes The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the probability that a flight is more than 140 minutes? A. 1.00 B. 0.5 C. 0.333 minutes D. 10 minutes For a standard normal distribution, what is the probability that z is greater than 1.75? A. 0.0401 B. 0.0459 C. 0.4599 D. 0.9599 What is the area under the normal curve between z = -1.0 and z = -2.0? A. 0.0228 B. 0.3413 C. 0.1359 D. 0.4772 The mean amount spent by a family of four on food per month is $500 with a standard deviation of $75. Assuming that the food costs are normally distributed, what is the probability that a family spends less than $410 per month? A. 0.2158 B. 0.8750 C. 0.0362 D. 0.1151 The standard normal probability distribution is unique because it has: A. A mean of 1 and any standard deviation B. Any mean and a standard deviation of 1 C. A mean of 0 and any standard deviation D. A mean of 0 and a standard deviation of 1 A binomial distribution has 100 trials (n = 100) with a probability of success of 0.25 (p=0.25). We would like to find the probability of 34 or more successes using the normal distribution to approximate the binomial. What z-score should be used?
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