00:01
Hello students, to find the maximum likelihood estimate of the parameter theta, we need to determine the value of theta that maximizes the likelihood function.
00:11
The likelihood function is the product of the probabilities of the observed values.
00:16
Here, l of theta is equal to p of x is equal to 0 into p of x is equal to 0 into p of x is equal to 1 into p of x is equal to 1 into p of x is equal to 1 into p of x is equal to 2 into p of x is equal to 2 into p of x is equal to 2 into p of x is equal to 3 into p of x is equal to 3.
00:49
We can simplify this expression using the given probability mass function.
00:54
Therefore, l of theta is equal to 2 theta divided by 3 whole square into theta by 3 whole cube into 2 into 1 minus theta divided by 3 whole cube into 1 minus theta divided by 3 whole square.
01:19
Now, l of theta can be simplified as 2 raised to 5 divided by 3 raised to 10 into theta raised to 5 into 1 minus theta raised to 5.
01:34
Taking logarithm of the likelihood function, we will get it as log l theta is equal to 5 log 2 minus 10 log 3 plus 5 log theta plus 5 log 1 minus theta.
02:05
To find the maximum likelihood estimate, we differentiate the log likelihood function with respect to theta and set it equal to 0.
02:14
Therefore, d log l theta with respect to theta is 5 log 2 differentiation is 0 minus 10 log 3 with respect to theta its differentiation is 0.
02:33
Taking 5 constant, derivative of log theta is 1 by theta...