# Finding the Maximum Likelihood Estimate for Theta in a Random Sample of Size 8.

• semidevil
In summary, the conversation is about finding the maximum likelihood estimate for the parameter theta given a random sample of size 8 with specific values for x1-x8. The first step is to find the likelihood function, which involves taking the product of the given function and then finding the natural log and differentiating it. However, there is confusion about which variable is the parameter and the lack of mention of a sample. A reminder is given that the maximum likelihood estimate is the value of the parameter that makes the probability of getting that particular sample the largest.
semidevil
I have a big test coming up, regarding estimators, but I just can't figure out the basics of maximum likilieehood.

so given this example, is this right?

p(k;theta) = theta^k * (1 - theta)^(1 - k), k = 0 1, and 0 < theta < 1.

so it's just the product of the function, and I get:

theta^K * 1 - (theta)^(sum from 1 to n of (k - n)).

then I take the natural log, and differentiate it to get

k/theta + (k - n)/ (1 - theta) = 0.

now, all I need to do is to put it in terms of theta...

first, did I do the first part right? in terms of finding the likelihoo function?

there's a lot of variables and I get confused when I do the products

It's hard to say- you haven't told us what n is or what k and theta are. Which is the parameter? Or are you doing a maximum likelihood estimate for both parameters at the same time?

In particular, I don't see any mention of a SAMPLE. The maximum likelyhood estimate for a parameter(s), given a SAMPLE, is the value of the parameter(s) that makes the probability of getting that particular sample largest.

HallsofIvy said:
It's hard to say- you haven't told us what n is or what k and theta are. Which is the parameter? Or are you doing a maximum likelihood estimate for both parameters at the same time?

In particular, I don't see any mention of a SAMPLE. The maximum likelyhood estimate for a parameter(s), given a SAMPLE, is the value of the parameter(s) that makes the probability of getting that particular sample largest.

sorry...ok, so this is a random sample of size 8 w/ x1 = 1, x2 = 0, x3 = 1, x4 = 1, x5 = 0, x6 =1, x7 = 1, and x8 = 0.

I need to find the MLE for theta.

## 1. What is MLE and why is it important in scientific research?

The maximum likelihood estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution based on a set of observed data. It is important in scientific research because it allows us to make inferences about the underlying population based on a sample of data. MLE is widely used in various fields such as biology, economics, and psychology.

## 2. How do you calculate the MLE?

The MLE is calculated by finding the values of the parameters that maximize the likelihood function, which is a measure of how likely the observed data is to occur given the specific values of the parameters. This can be done analytically or numerically using various optimization algorithms.

## 3. What assumptions are made when using MLE?

MLE assumes that the data follows a specific probability distribution and that the observations are independent and identically distributed. It also assumes that the data is complete, meaning there are no missing values.

## 4. Can MLE be used for any type of data?

MLE can be used for any type of data as long as the data follows a specific probability distribution. If the data does not fit a known distribution, other methods such as the method of moments or Bayesian estimation may be more suitable.

## 5. How do you know if the MLE is a good fit for your data?

There are various goodness-of-fit tests that can be used to assess whether the MLE is a good fit for the data. These include the Kolmogorov-Smirnov test and the chi-square test. Additionally, visual inspection of the data and the fitted distribution can also provide valuable insight into the adequacy of the MLE.

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