2. You have two lightbulbs for a particular lamp. Let X = the lifetime of the first bulb and Y = the lifetime of thesecond bulb (both in 1000s of hours). Suppose that X and Y are independent and that each has an exponentialdistribution with parameter λ=1• What is the joint pdf of X and Y?• What is the probability that each bulb lasts at most 1000 hours (i.e., X ≤1 and Y ≤1)?

X and Y have exponentital distribution with parameters want to calculate the joint PDF of X and…

Answered: 2. You have two lightbulbs for a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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The time, T (measured in seconds), between successive hits on a website (“inter-hit time” in the rest of the problem) follows the exponential distribution with parameter 1/2. (a) What is the probability that a randomly selected inter-hit time is between 1 and 3 seconds? (b) Assume that different inter-hit times are independent of each other. A random sample of 810 inter-hit times is obtained. Give an exact symbolic expression for the probability that more than 290 of the inter-hit times in the sample are between 1 and 3 seconds? (c) Obtain a good numerical approximation for the probability from part (b)
Cardiovascular Disease Suppose the incidence rate of myocardial infarction (MI) was 5 per 1000 among 45- to 54-year-old men in 2000. To look at changes in incidence over time, 5000 men in this age group were followed for 1 year starting in 2010. Fifteen new cases of MI were found. Suppose that 25% of patients with Mi in 2000 died within 24 hours. This propartion is called the 24-hour case-fatality rate. Of the 15 new MI cases in the preceding study. 5 died within 24 hours. Test whether the 24-hour case- fatality rate changed from 2000 to 2010.
Suppose model (XY, XZ, YZ) holds in a 2 x 2 x 2 table, and the common XY conditional log odds ratio at the two levels of Z is positive If the XY and YZ conditional log odds ratios are both positive or both negative, show that the XY marginal odds ratio is larger than the XY conditional odds ratio.
Let X equal the IQ of a randomly selected American. Assume X ~ N( μ μ {"version":"1.1","math":"μ"} =100, σ σ {"version":"1.1","math":"σ"} =4). What is the probability that a randomly selected American has an IQ below 90?
Suppose the lifespan (in months) of a smartphone battery can be modeled as a continuous random variable with CDF F(x) = 1 − e-x/3 x ≥ 0 What is the probability that the battery lasts between 12 to 15 months?
Let X be a normal random variable with mean 85 and a variance of 25 (i.e., X ∼ N (85, 25)). step1: Let M = aX + b for some constants a, b not equal to 0. Write an expression for the pdf of M . step 2: Suppose that X represents an approximate distribution of the final scores in a certain math course (ignore the fact that this approximation can technically have scores that are greater than 100 or less than 0). If the teacher were to curve the scores, it means he would determine a function to apply to the scores to achieve a desired distribution (assume an affine function in this case, as in the previous part). Suppose he wants the scores to be normally distributed with mean 80 and a standard deviation of 4. What should he choose for a and b? What students would see their score lowered, and what students would see their score increased?
The Poisson distribution gives the probability for the number of occurrences for a "rare" event. Now, let x be a random variable that represents the waiting time between rare events. Using some mathematics, it can be shown that x has an exponential distribution. Let x > 0 be a random variable and let β > 0 be a constant. Then y = 1 β e−x/β is a curve representing the exponential distribution. Areas under this curve give us exponential probabilities. If a and b are any numbers such that 0 < a < b, then using some extra mathematics, it can be shown that the area under the curve above the interval [a, b] is the following. P(a < x < b) = e−a/β − e−b/β Notice that by definition, x cannot be negative, so, P(x < 0) = 0. The random variable x is called an exponential random variable. Using some more mathematics, it can be shown that the mean and standard deviation of x are the following. μ = β and σ = β Note: The number e = 2.71828 is used throughout…
Let L be the random variable for the length of time, in years, that a person will remember an actuarial statistic. For a certain popula- tion, L is exponentially distributed with mean 1/Y, where Y has a gamma distribution with a = 4.5 and 3 = 4. Find the variance of L.
Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample?
Suppose that Y has an exponential distribution with mean Beta. Show that 2Y/Beta has a chi square distribution with 2 degrees of freedom.
The life X (in hours) of a battery in constant use is a random variable with exponential density. What is the probability that the battery will last more than 12 hours (h) if the average life is 8 h?
Question 7 According to the data, the weight of a randomly selected checked-in luggage has a normal distribution with a mean of 51 lbs and a standard deviation of 9 lbs. Let X be the weight of a randomly selected checked-in luggage and let S be the total weight of a random sample of size 33. 1. Describe the probability distribution of X and state its parameters and o: X~ Select an answer (μ = Select an answer unknown X² B N T 2. Use the Central Limit Theorem and find the probability that the weight of a randomly selected checked-in luggage is less than 61 lbs. (Round the answer to 4 decimal places) Check-In Select an answer Select an answer the original population is normally distributed the sample size is large (n>30) although the distribution of the original population is unknown the sample size is small (n<30) and the distribution of the original population is unknown the distribution of the original population is unknown S~ Select an answer (s= Select an answer σ= to describe the…

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