Exercise 16. The x²(v) distribution is a special case of Gamma distribution (not to be con-fused with gamma function; see below). The density function of the Gamma distribution withparameters and k is given bywhere4(x) ==1r(k) okpk-1e2if x > 0, andotherwise,1.-xa*-¹e-² dxr(k) =is the gamma function. For every k ≥ 1, 0 > 0, find the point at which y(x) has its maximum.Hints: The algebra can be simplified by appropriate use of logarithms.

Given the density function of the Gamma distribution with parameters θ and k as φx=1Γkθkxk-1e-xθ, if…

Answered: Exercise 16. The x²(v) distribution is…

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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(c) Let X follow the gamma distribution with pdf S(z) = r(a) on z> 0. If a =1 and V(X)=4 find the value of b, and hence find E(X).
For what value of c is the function f(x) = {c/x³ if x ≥ 1 {gx3 otherwise a probability density function?
Consider a group of people who have received treatment for a disease such as cancer. Let t be the survival time, the number of years a person lives after receiving treatment. The density function giving the distribution of t is p(t) = Ce-Ct for some positive constant C, and the cumulative distribution function is P(t) = , p(x)dx. Think carefully about what the practical meaning of P(t) is, being sure that you can put it into words. a) The survival function, S(t), is the probability that a randomly selected person survives for at least t years. Find a formula for S(t). b) Suppose that a patient has a 80 percent chance of surviving at least 2 years. Find C. c) Using the value of C you found in (b), find the probability that the patient survives up to (that is, less than or equal to) 1 years. d) Using the value of C you found in (b), find the the mean survival time for patients with this survival function, in years.
The normal probability density function g(x) = ov2n Where u = mean, o = standard deviation, n = 3.14159, e 2.71828 Let's study a special case where u = 0 and o = 1, which indicates a standard normal probability density function. For simplicity, we scale the function so as to remove the factor 1/(ov2n). Now, we have a new function f(x) f(x) = e (a) Find f'(x) and show the intervals of increase or decrease for f. (b) Find the extreme value of f. (c) Find the intervals of concavity and the inflection points. (d) Sketch the graph of f and point out the extremum value and inflection points.
(b) The probability density of a function is given by p(x) from x = 1 to x = 4. That is 62 x3 dx = 1 62 P(x) dx = - 00 Determine the value of the standard deviation.
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…
Consider times at which a parent provisions nestlings, that is, the time be- tween arrivals of a parent carrying food for the nestlings. Suppose that the waiting time between arrivals of a parent bird follows the exponential density function f (t) = 0.6e-0,61 for t z 0 and f (t) = 0 for t < 0, where the waiting times are in minutes. (a) What is the mean waiting time? (b) Find the probability that the wait will be between 2 and 5 minutes. (c) Find the probability that the wait will be at least 3 minutes.
7. Find the related density function for the following distribution functions: 1 (a) F(x)= arctan x, − or co T I (b) F(x)=0 if x <0, 1-e-*if x≥0; 0≥0. 2 x² (c) F(x) = 0, if x < 0, + if 0 < x < 1, 2x - 2 2 1, if 1
The exponential density function is given by if t 0 where u is the mean. Use it to answer the question below. (a) A lamp has two bulbs of a type with an average lifetime of 1000 hours. Assuming that we can model the probability of failure of these bulbs by an exponential density function with mean µ = that both of the lamp's bulbs fail within 1000 hours. (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 1000 hours. 1000, find the probability

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