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 bywherey(x) =1T(k) ok--¹e-/, if x > 0, andotherwise,T(k) = √√pk-le-dis the gamma function. For every k ≥ 1, 0 > 0, find the point at which p(x) has its maximum.Hints: The algebra can be simplified by appropriate use of logarithms.

Introduction: The Gamma distribution is a family of the continuous probability distribution that is…

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

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A continuous random variable has the probability density function of f(x)=0.0118x and the interval of [O 13]. What is the probability that this variable has a value between 1 and 2? Answer: A continuous random variable has the probability density function of f(x)=0.0118x and the interval of [O 13]. What is the cumulative distribution function for the value of 3? Answer:
At a restaurant, the density function for the time a customer has to wait before being seated is given by f(t) = 2e¯ if t > 0. Find the probability that a customer will have to wait at least 6 minutes for a table. Probability = .000006 -2t
(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?
6. Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-x, o0) is a function f such that f(r) 20 and (2) = = 1. (a) Determine which of the following functions are probability density functions on the (-x0, 00). fr-1 00 (b) We can also use probability density functions to find the erpected value of the outcomes of the event - if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. rf(x) dr yields the expected value for a density f(r) with domain on the real numbers.) Find the expected value for one of the valid probability densities above.
(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.
13 Let X be the standard normal distribution and consider the transformation given by Y = 1/X. Select all answers that apply. Select all answers that apply. Note exp(z) = e^z %D %3D Choose all that apply. O The density of Y is: g(y) = (1/2*pi)^(1/2) * (1/y^2)*exp(-1/(2y^2)) The density of Y is: g(y) = (1/2*pi)^(1/2) *exp(-1/(2y^2)) The values of y for which the density is non zero is y0. The values of y for which the density is non zero is y>0. %3D
Let X have a gamma distribution with pdf 1 f(x) Γ(α)βα What is the probability density function of Y = ex? ·xα-¹e-x/B, 0 0, ß > 0
Q.3 The probability density function for a continuous random variable X is fx(x) = {a + bx²; 0
he length of time, in minutes, for an airplane to obtain clearance for takeoff at a certain airport is a random variable Y = 3X-2, where X has the density function. Find the mean and variance of the random variable Y.
Q.1

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