Consider a random sample n from a population

In summary, the maximum likelihood estimator for parameter p in a random sample from a population with probability distribution f(x,p) depends on p is given by the likelihood function L(p|x) = p^x (1-p)^1-x. This function can be broken into two factors, one for x=0 and the other for x=1, to better understand how the products look.
  • #1
TomJerry
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0
Problem: Consider a random sample n from a population with probability distribution f(x,p) that depends on parameter p. Find the maximum likelihood estimator for p when

f(x,p) = p^x (1-p)^1-x for x=0,1
 
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  • #2
So you've tried...?
 
  • #3
statdad said:
So you've tried...?

I m having difficulty starting , can you show me an example which is near to this or related to this . I just need a starting point .
 
  • #4
The likelihood function is

[tex]
L(p \colon x_1, x_2, \dots, x_n) = \prod_{i=1}^n p^{x_i} (1-p)^{1-x_i}
[/tex]

Break the product into two factors, one in which [tex] x_j = 0 [/tex], the other in which [tex] x_j = 1 [/tex], and see what the products look like.
 
  • #5


The maximum likelihood estimator for p can be found by maximizing the likelihood function, which is the product of the probabilities of obtaining the observed sample values. In this case, the likelihood function is:

L(p) = p^x (1-p)^1-x

To find the maximum likelihood estimator, we take the derivative of the likelihood function with respect to p and set it equal to 0:

dL/dp = x*p^(x-1)*(1-p)^1-x - (1-x)*p^x*(1-p)^-x = 0

Simplifying this equation, we get:

p^(x-1)*(1-p)^-x = (1-x)*p^x*(1-p)^1-x

Dividing both sides by p^x*(1-p)^1-x, we get:

p^-1*(1-p)^-1 = 1-x

Rearranging this equation, we get:

p*(1-p) = x

This is a quadratic equation with two solutions for p:

p = (1 ± √(1-4x))/2

However, we must also consider the constraint that p must be between 0 and 1. Therefore, the maximum likelihood estimator for p is:

p = (1-√(1-4x))/2 for x < 1/4

p = (1+√(1-4x))/2 for x ≥ 1/4

In other words, the maximum likelihood estimator for p is the smaller root of the quadratic equation if x is less than 1/4, and the larger root if x is greater than or equal to 1/4. This estimator will provide the most likely value for p based on the observed sample values, and it is known to be efficient and consistent in estimating the true value of p.
 

What is a random sample?

A random sample is a subset of a larger population that is chosen in a way that every member of the population has an equal chance of being included in the sample. This ensures that the sample is representative of the entire population.

Why is a random sample important?

A random sample is important because it allows for generalizations to be made about the entire population based on the characteristics observed in the sample. This is useful in scientific research as it can save time and resources by not having to collect data from every single member of the population.

How is a random sample selected?

A random sample can be selected using various methods such as simple random sampling, systematic sampling, stratified sampling, or cluster sampling. In simple random sampling, each member of the population has an equal chance of being chosen. In systematic sampling, a starting point is randomly selected and then every nth member is chosen. Stratified sampling involves dividing the population into subgroups and then randomly selecting from each subgroup. Cluster sampling involves randomly selecting clusters or groups from the population.

What is the sample size and why is it important?

The sample size refers to the number of individuals included in the random sample. It is important because it can affect the accuracy and precision of the results. A larger sample size can provide more reliable results, while a smaller sample size may not be representative of the entire population.

What are the advantages and disadvantages of using a random sample?

The advantages of using a random sample include its ability to provide results that can be generalized to the entire population, its cost-effectiveness, and its unbiased nature. However, a random sample may not always accurately represent the population and may not be feasible in certain situations where the population is too large or dispersed.

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