Using sampling if a single value is estimated for the unknown parameter of the population, then this process of estimation is called point estimation. We shall discuss two methods of point estimation below:

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**I. Method of Maximum Likelihood**

Let *x _{1},x_{2},…, xn *be a random sample from a population whose

*p.mf*(discrete case) or

*p.df*

(continuous case ) is *f (x, θ*) where *θ* is the parameter. Then construct the likelihood function as follows :

L = *f *(x_{1, }*θ) . f (x2 , θ) … f (xn, θ)*

Since log L is maximum when L is maximum. Therefore to obtain the estimate of *θ*, we maximize L as follows,

Here *θ* is called Maximum Likelihood Estimator (MLE).

**Properties of MLE**

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- MLE is not necessarily unbiased.
- MLE is consistent, most efficient and also sufficient, provided a sufficient estimator exists.
- MLE tends to be distributed normally for large samples.
- If g(
*θ*) is a function of*θ*and e is an MLE of*θ*, then g(*θ*) is the MLE of g(*θ*).

**Example 4. ***A discrete random variable X can take up all non-negative integers and *

* *

*P (X= r) *= *p (1- p) ^{r} (r *=

*0, 1, 2, .. .)*

*here, p (0 *< *p *< *I) is the parameter of the distribution. Find the MLE of p for a sample of size: x _{1},x_{2},…, xn from the population of X*

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**Solution: **Consider the following likelihood function:

Taking log on both sides we obtain

Hence the MLE of *p *is 1/ 1+x

**Example 5.** A random variable X has a distribution with density function :

*f* (x) = λ x ^{λ -1}(0 < x < 1)

where_ A. is the parameter. Find the MLE of 'A for a sample of size n. *x _{1},x_{2},…, xn*, from the population of X.

**Solution.** Consider the following likelihood function :

Taking log on both sides we obtain

ln L = n in A. + (A.- 1) In (*x _{1},x_{2},…, xn)*

Taking log on both sides we obtain

ln L = n in A. + (A.- 1) In (*x _{1},x_{2},…, xn)*

**Example 6.** X tossed a biased coin 40 times and got head 15 times, while Y tossed it 50 times and got head 30 times. Find the MLE of the probability of getting head when the coin is tossed.

**Solution.** Let P be the unknown probability of getting a head.

Using binomial distribution,

Probability of getting 15 heads in 40 tosses Probability of getting 30 heads in 50 tosses The likelihood function is taken by multiplying these probabilities.

=> P= ½,which is the MLE

**II. Method of Moments**

In this method, the first few moments of the population is equated with the corresponding moments of the sample.

Then µ’_{r }= m’_{r}

Where

=> P= ½,which is the MLE

**II. Method of Moments**

In this method, the first few moments of the population is equated with the corresponding moments of the sample.

Then µ’_{r }= m’_{r}

The solution for the parameters gives the estimates. But this method is applicable only when population moments exist.

**Example 7.** Estimate the parameter p of the binomial distribution by the method of moments when n is known).

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which is the estimated value.