**Proof of the Variance of a Beta-Distributed Random Variable**The variance of a beta-distributed random variable with parameters \(\alpha\) and \(\beta\) is given by:\[\sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\]To prove this formula, one must derive the expression starting from the definition and properties of the beta distribution. Here, \(\alpha\) and \(\beta\) are shape parameters of the distribution.

Answered: Image /qna-images/answer/b9a45894-a1eb-4873-aff0-9c9fa449052d.jpg

Answered: Prove that the variance of a…

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...

See similar textbooks

Similar questions

X is a discrete random variable such that the possible values are {a, a+1, a+2, ... b-1, b}. The random variable has a mean of 7 and a variance of 4. What is the Pr(X<=6|X>4)?
For a random variable X, suppose that E[X] = 1 and Var(X) = 5. Then %3D (a) E[(2 + X)²] = (b) Var(2 + 3X) = Netsu rtiol orodit on this problem
Let the random variable X be defined on the support set (1,2) with pdf fX(x) = (4/15)x3, Find the variance of X.
Imagine a family member looks over your shoulder as you look at the variance equation Σki=1(xi − μ)2P(X = xi) and asks why the P(X = xi) term is there. What would you say?
Suppose that X₁,..., Xn are i.i.d. from a normal population with mean μ and variance o². Calculate the expected value and the variance of the sample mean X and find the distribution of √n(X − µ).
Let X11 X12, Xini and X21, X22, X2n2 be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = = ². Suggest an unbiased estimator of o². |
In a large lot of manufactured items, 10% contain exactly one defect and 5% contain more than one defect. If ten items are randomly selected from this lot for sale, the repair cost total Y1 +3Y2, where Y1 denote the number among the ten having one defective and let Y2 denote the number with two or more defects. Find the expected value and variance of the repair cost. Assume that Cov(Y1, Y2) =-0.05.
Suppose that Y₁ is a negative binomial random variable with p= .75 and r = 6, Y₂ is a random variable with mean -5 and variance 11, and Y3 is a gamma random variable with 1.6 and a = 5.9. Further suppose that all of these random variables are independent. (a) What is E[2Y₁ − 6Y2 + 5Y₁Y3]? (b) What is the variance of 3Y₁ +5Y2 - 8Y3? (c) What is the mgf of 7Y₁ +4Y3? 0 =

Recommended textbooks for you

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

SEE MORE TEXTBOOKS

Prove that the variance of a beta-distributed random variable with parameters a and 3 is αβ (a + B)² (a + B + 1)* =