Solution Found!
Let X ~ Bin(n, p).a. Show that if x is an integer between
Chapter 4, Problem 28SE(choose chapter or problem)
Let \(X \sim \operatorname{Bin}(n, p)\).
a. Show that if x is an integer between 1 and n inclusive, then
$$\frac{P(X=x)}{P(X=x-1)}=\left(\frac{n-x+1}{x}\right)\left(\frac{p}{1-p}\right)$$
b. Show that if \(X \sim \operatorname{Bin}(n, p)\), the most probable value for X is the greatest integer less than or equal to\(n p+p\) . [Hint: Use part (a) to show that \(P(X=x) \geq P(X=x-1)\) if and only if \(x \leq n p+p\).]
Equation Transcription:
.
Text Transcription:
X Tilde Bin(n,p)
P(X=x)/P(X=x-1) = (n-x+1/x) (p/1-p)
np + p
P(X=x) greater than or equal to P (X=x-1)
x less than or equal to np + p
Questions & Answers
QUESTION:
Let \(X \sim \operatorname{Bin}(n, p)\).
a. Show that if x is an integer between 1 and n inclusive, then
$$\frac{P(X=x)}{P(X=x-1)}=\left(\frac{n-x+1}{x}\right)\left(\frac{p}{1-p}\right)$$
b. Show that if \(X \sim \operatorname{Bin}(n, p)\), the most probable value for X is the greatest integer less than or equal to\(n p+p\) . [Hint: Use part (a) to show that \(P(X=x) \geq P(X=x-1)\) if and only if \(x \leq n p+p\).]
Equation Transcription:
.
Text Transcription:
X Tilde Bin(n,p)
P(X=x)/P(X=x-1) = (n-x+1/x) (p/1-p)
np + p
P(X=x) greater than or equal to P (X=x-1)
x less than or equal to np + p
ANSWER:
Answer:
Step 1 of 2:
(a)
In this question, we a asked to prove the following.
…………(1)
Where x is an integer between 1 and n inclusive and is probability.
If is a random variable whose distribution is binomial with parameter and , then we can write
The probability mass function of a binomial random variable is defined by,
= otherwise
………………..(2)
Replace with in equation (2), we get
………………….(3)
=
(can be written as
can be written as
simplify the equation
=
= (
= (
Hence proved.