Roughly, speaking, we can use probability density functions to model the likelihood of anevent occurring. Formally, a probability density function on (-∞,0) is a function f suchthatf(2) 20andLsla) = 1.(a) Determine which of the following functions are probability density functions on the(-00, 00).(1-1 00(b) We can also use probability density functions to find the expected value of the outcomesof the event - if we repeated a probability experiment many times, the expected valuewill equal the average of the outcomes of the experiment. (e.g. zf(z) dz yields theexpected value for a density f(x) with domain on the real mumbers.) Find the expectedvalue for one of the valid probability densities above.

Answered: Image /qna-images/answer/6b78e19d-61eb-481d-9c03-0176c6f2e2b9.jpg

Answered: Roughly, speaking, we can use…

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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Suppose a college professor never finishes her lecture before the end of the class period, and always finishes within five minutes after the class period is supposed to end. Let X = time that elapses between the end of the class period and the actual end of the lecture. Suppose the pdf of X is: (image) Find the value of k that makes f(x) a legitimate probability density function, and use that value of k to find the probability that the lecture ends less than 3 minutes after the class period is supposed to end.
Suppose a random variable has a continuous uniform distribution between 0 and 10, such that its probability density function is: f(x) = 1/10. a) According to Chebychef’s rule, what is the smallest probability a random x will within 2 standard deviations of its mean? i.e. P(|x − µ| <2 ∗ σ) b) What is the exact probability that a random x will fall within 2 standard deviations of its mean for this uniform distribution?
Found out the join probability density function, not too sure if thats correct.
The probability density function of the continuous random variable X defined in the set of non-negative real numbers is given as f(x) = 2.exp(-2x). What is the expected value of X?
The probability density function of a random variable X is given by Find the probability that it will take on a value within two standard deviations of the mean andcompare this probability with the lower bound provided by Chebyshev’s theorem.
Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.
If the random variable T is the time to failure of a commercial product and the values of its probability den-sity and distribution function at time t are f(t) and F(t), then its failure rate at time t is given by f(t)1 − F(t). Thus, thefailure rate at time t is the probability density of failure attime t given that failure does not occur prior to time t.(a) Show that if T has an exponential distribution, thefailure rate is constant. (b) Show that if T has a Weibull distribution (see Exer-cise 23), the failure rate is given by αβt β−1.
Suppose that the probability density function of the length of computer cables is f (x) = 2x/(32) for x between 0 and 3 meters. Determine the mean of the cable length. Please enter the answer to 2 decimal places.
First a point is X selected at random from the interval (0,1). Then another point is selected at random from the interval (x,1) in such a way so that x+y>=1. Find the probability density function of y
Suppose an electric-vehicle manufacturing company estimates that a driver who commutes 50 miles per day in a particular vehicle will require a nightly charge time of around 1 hour and 30 minutes (90 minutes) to recharge the vehicle's battery. Assume that the actual recharging time required is uniformly distributed between 70 and 110 minutes. (a) Give a mathematical expression for the probability density function of battery recharging time for this scenario. f(x) = , 70 ≤ x ≤ 110 , elsewhere
Suppose that the lifetime X (in hours) of a certain type of flashlight battery is a random variable on the interval 30 ≤ x ≤ 50 with density function f(x) = 1/20, 30 ≤ x ≤ 50. Find the probability that a battery selected at random will last at least 35 hours.
(a) A lamp has two bulbs, each of a type with average lifetime 1800 hours. Assuming that we can model the probability of failure of these bulbs by an exponential density function with mean μ = 1800, find the probability that both of the lamp's bulbs fail within 1800 hours. (Round your answer to four decimal places.)(b) Another lamp has just one bulb of the same type as in part (a). If one bulb burns out and is replaced by a bulb of the same type, find the probability that the two bulbs fail within a total of 1800 hours. (Round your answer to four decimal places.)

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