Let $$x_1, x_2, $$ be uniformly distributed on the interval 0 to a. recall that the maximum likelihood estimator of a is $$a = max(x_i)$$. argue intuitively why aˆ cannot be an unbiased estimator for a. b. suppose that e(a) = na/(n + 1). is it reasonable that aˆ consistently underestimates a? show that the bias in the estimator approaches zero as n gets large. c. propose an unbiased estimator for a. d. let $$y = max(x_i)$$. use the fact that y ≤ y if and only if each $$x_i ≤ y$$ to derive the cumulative distribution function of y . then show that the probability density function of y is. $$f(y) = [ny^n - ^1/a^n 0$$, 0 ≤ y ≤ a otherwise, use this result to show that the maximum likelihood estimator for a is biased. e. we have two unbiased estimators for a: the moment estimator $$a_1=2\overline{\mbox{x}}$$ and $$a_2 = [(n + 1)/n] max(x_i)$$, where max $$(x_i)$$ is the largest observation in a random sample of size n. it can be shown that $$v(a_1) = a^2/(3n)$$ and that $$v(a_2) = a^2/[n(n + 2)]$$. show that if n > 1, aˆ2 is a better estimator than aˆ. in what sense is it a better estimator of a?

a)  

For this case the value for is always smaller than the value of a, assuming So then for this case it cannot be unbiased because an unbiased estimator satisfy this property:

and that's not our case.

b)

Since is a negative value we can conclude that underestimate the real value a.

c)

And assuming independence we have this:

e) On this case we see that the estimator is better than and the reason why is because:

and that's satisfied for n>1.

Step-by-step explanation:

Part a

For this case we are assuming

And we are are ssuming the following estimator:

For this case the value for is always smaller than the value of a, assuming So then for this case it cannot be unbiased because an unbiased estimator satisfy this property:

and that's not our case.

Part b

For this case we assume that the estimator is given by:

And using the definition of bias we have this:

Since is a negative value we can conclude that underestimate the real value a.

And when we take the limit when n tend to infinity we got that the bias tend to 0.

Part c

For this case we the followng random variable and we can find the cumulative distribution function like this:

And assuming independence we have this:

Since all the random variables have the same distribution.  

Now we can find the density function derivating the distribution function like this:

Now we can find the expected value for the random variable Y and we got this:

And the bias is given by:

And again since the bias is not 0 we have a biased estimator.

Part e

For this case we have two estimators with the following variances:

On this case we see that the estimator is better than and the reason why is because:

and that's satisfied for n>1.

36 girls

step-by-step explanation:

the 48 boys is 4 parts of the ratio

divide 48 by 4 to obtain one part of the ratio

= 12 ← 1 part of the ratio

3 parts = 3 × 12 = 36 ← number of girls