Roughly, speaking, we can use probability density functions to model the likelihood of anevent occurring. Formally, a probability density function on (-0, 0) is a function ƒ suchthatf(x) > 0and| f(x) = 1.(a) Determine which of the following functions are probability density functions on the(-0, 00).0 < x < e(i) f(x) =otherwise-20 < x < 2/2(ii) ƒ(x) =(x – /2)3otherwisededx 0 0(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. L xf(x) dx yields theexpected value for a density f(x) with domain on the real numbers.) Find the expectedvalue for one of the valid probability densities above.

Answered: Image /qna-images/answer/398673ac-972f-4af6-a128-a5321c2b02d4.jpg

Answered: Roughly, speaking, we can use…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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The exponential distribution is a probability distribution for non-negative real numbers. It is often used to model waiting or survival times. The version that we will look at has a probability density function of the form p(y|v) = exp(-e¯ºy — v) (1) where y R+, i.e., y can take on the values of non-negative real numbers. In this form it has one parameter: a log-scale parameter v. If a random variable follows an exponential distribution with log-scale v we say that Y~ Exp(v). If Y~ Exp(v), then E [Y] =e" and V [Y] = e²v. 1. Produce a plot of the exponential probability density function (1) for the values y E (0, 10), for v = 1, v = 0.5 and v= 2. Ensure the graph is readable, the axis are labeled appropriately and a legend is included. ta il 2. Imagine we are given a sample of n observations y = (y₁,..., yn). Write down the joint proba- bility of this sample of data, under the assumption that it came from an exponential distribution with log-scale parameter v (i.e., write down the…
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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.
"Time headway" in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let X = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow. The pdf of X is the following. 0.17e-0.17(x - 0.5) - {0.17e-0.1 x ≥ 0.5 otherwise (a) What is the probability that time headway is at most 6 sec? (Round your answer to three decimal places.) f(x) = (b) What is the probability that time headway is more than 6 sec? At least 6 sec? (Round your answers to three decimal places.) more than 6 sec at least 6 sec (c) What is the probability that time headway is between 5 and 6 sec? (Round your answer to three decimal places.)
Suppose that each individual in a large insurance portfolio incurs losses according to an exponential distribution with mean 1/2,where i varies over the portfolio according to a G(a, 3) mixing distribution. The respective densities of the two distributions are given by Sx (x) = (1/2) exp (-x/à), x > 0, 2 > 0; S(2) =- T(a) 2-l exp(-8i), 2 > 0. Given that the Pareto pdf given by fx (x) = (5 +x)ª+1 * > 0, a > 0, 8> 0. (a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(a, 5). (b) Use the mixing formulation of the Pareto to deduce that if X~P(a, 5), then E (X) = .

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