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Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 4.9 - Problem 7e
Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 4.9 - Problem 7e

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# Maximum likelihood estimates possess the property of ISBN: 9780073401331 38

## Solution for problem 7E Chapter 4.9

Statistics for Engineers and Scientists | 4th Edition

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Problem 7E

Maximum likelihood estimates possess the property of functional invariance, which means that if is the MLE of θ, and h is any function of θ, then h (θ) is the MLE of h(θ).

a. Let .X ~ Bin(n, p) where n is known and p is unknown. Find the MLE of the odds ratio p/(1- p).

b. Use the result of Exercise 5 to find the MLE of the odds ratio p/(1 - p) if X ~ Geom(p).

c. If X ~ Poisson(λ), then P(X = 0) = e-λ. Use the result of Exercise 6 to find the MLE of P(X = 0) if X1,…,Xn is a random sample from a population with the-Poisson(λ) distribution.

Step-by-Step Solution:

Step 1 of 3:

Let, be the maximum likelihood estimator for and h( ) be the MLE of h( ).

Let X follows Binomial distribution with probability mass function

f(x;p) =  (1-p The claim is to find the MLE of the odds ratio The likelihood function is

L(x:p) =  (1-p Take log on both sides

Log L(x:p) = Log( )

= [ log(n!) - log(x!) - log(n-x)! + x log p + (n - x) log (1-p) ]

Differentiate with respect to p and equate it to zero. (log(n!) - log(x!) - log(n-x)! + x log p + (n - x) log (1-p) ) = 0

= (x log p + (n - x) log (1-p) )

Where, (x log p) = and (n - x) log (1-p) = Substitute these values to (x log p + (n - x) log (1-p) )

Therefore, (x log p + (n - x) log (1-p) ) = - Then, = For = = = Hence, MLE of the odds ratio is Step 2 of 3

Step 3 of 3

##### ISBN: 9780073401331

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