Problem 2 let x1 x2 xn be a sample data and let c be a non...

Problem 2) Let $x_1, x_2, ..., x_n$ be a sample data and let $c$ be a non-zero constant. a) For what values of $c$ is the quantity $\sum(x_i - c)^2$ minimized? (Hint: Take the derivative with respect to $c$, set equal to zero, and solve.) b) Suppose that $\mu$ is the mean of the population from where the above sample has been collected. Explain which quantity is smaller: $\sum(x_i - \bar{x})^2$ or $\sum(x_i - \mu)^2$? Assume that $\mu \ne \bar{x}$.

Transcript

00:01 In this video, we are going to focus on the given information that for a random sample of size n, from a population of size capital n, we need to consider the estimate of mu, and it is as follows, that x of c bar is equal to here summation i is equal to 1 and it will be going to n.

00:19 And now here we are having ci multiplied with xi.

00:23 So here, ci are the fixed constant and xi are the sample units.

00:27 So here in the first part of this very problem, we need to determine the condition on ci in such a way that estimate is unbiased.

00:35 So here we can say that e of x bar c is going to be here equals to.

00:42 So before moving ahead, we can here mention that an estimator is unbiased if its expectation is equal to the true population parameter value.

00:51 Thus, the estimator xc will be unbiased if its expectation is equal to mu.

00:57 So here we can check out the value that it is here basically c -i -x -i and now here we are going to be going to have i is equal to 1 and it is going up to n.

01:07 And this will be here summation, basically it is i is equal to 1 up to n and now it will be here e of ci times xi.

01:17 And thus if we simplify this very expression, we are going to have here the summation i is equal to 1 up to n and then here it is going to be.

01:27 Be ci and then it is here basically e of xi now we can say that this will be here equals to we can write this one as i is equal to 1 up to n and then here we are having ci and then it will be mu and this can be written because we know that the value of e of xi is equals to mu right so that's how we have written this very expression and now simplifying this very problem further we can say that it will be here equals to mu times the summation and then here it is i is equal to one and then it is going up to n and we are having here c i so this value is going to be here equals to one so if it is equal to one then we can here say that we are getting the value of mu to be equals to e of x of c and here if we if we have written this value as 1, then what does it mean? it means that the condition which is required is basically here we can say that condition on ci is equals to summation and then here it is i is equal to 1 and then it will be here n and then here we have ci and that is equal to 1...

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