2. Show a relation between the beta and the binomial distributions by proving the followingrelationr+レー1T(r +r +v – 1レー1P(X < p) =T(r)T(v) "-(1 - 2)v-ldx =I(r)I(v).j=rwhen r and v are integers.

Answered: 2. Show a relation between the beta and…

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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Question

2. Show a relation between the beta and the binomial distributions by proving the following
relation
r+レー1
T(r +
r +v – 1
レー1
P(X < p) =
T(r)T(v) "-(1 - 2)v-ldx =
I(r)I(v).
j=r
when r and v are integers.

Expert Solution

Step 1

Beta Distribution

Beta - Binomial distributions are specifically used for Bayesian models for developing marketing strategies and intelligence testing

A distribution if p is the probability of success and has shape parameters r>0 and v >0 then these parameters are represented as the probability of success.

Therefore, for large values, they tend to converge into binomial distribution

If m items are tested n times the binomial distribution formula is

$P (X = x_{i}) = {}_{n}{c_{x_{i}} \times p}^{x_{i}}_{i} \times {(1 - p_{i})}^{n - x_{i}}; i = 1, 2, 3, . . . n w h e r e x_{i} = t o t a l n u m b e r o f s u c c e s s e s p_{i} = p r o b a b i l i t y o f s u c c e s s n = n u m b e r o f t r a i l s$

Perhaps the probability density function of Beta distribution is given as: $f (x, r, v) = \frac{x^{r - 1} {(1 - x)}^{v - 1}}{B (r, v)} f (k | n, r, v) = \frac{Γ (n + 1)}{Γ (k + 1) Γ (n - k + 1)} \frac{Γ (k + r) Γ (n - k + v)}{Γ (n + r + v)} \frac{Γ (r + v)}{Γ (r) Γ (v)}$