Unbiased estimators of $k$ using two observations from $\mathsf{Unif}(0,k).$ [closed]

I basically do not know how to approach this question: Please let me know how to approach this question, and if you attach full explanation, I will appreciate it. Thanks.

• ad a) If I´m right $\mathbb E(X_1)=\mathbb E(X_2)=\frac{k+1}{2}$ and $\mathbb E(X_1+X_2)=k+1$ Thus $\mathbb E(X_1+X_2)\neq k$. Consequently $k$ is a $\color{blue}{\text{biased}}$ estimator. Feb 21 '18 at 16:32
• @callculus. It's a continuous RV. Feb 21 '18 at 17:59
• @BruceET Now I´m seeing it. Ouch. Feb 21 '18 at 18:06

In order to show that a function $f(x_1,x_2)$ is an unbiased estimator, all you have to do is to show that $E[f(X_1,X_2)] = k,$ where $k$ is the parameter. From your exercise we know that $X_1,X_2$ are uniformly distributed (we get that from their density function) so $E[X_1]=E[X_2] = k/2.$ The problem now is solved because: $E[f(X_1,X_2)] = E[X_1+X_2] = E[X_1] + E[X_2] = k.$

P.S : You should always write down your thoughts about the questions you post.

• Tidied up some formatting and punctuation--I hope without changing your meaning. Feb 21 '18 at 18:03
• Thanks a lot!!! Feb 21 '18 at 18:27

You already have a start on (a). Here is some guidance on other selected parts. Supply missing steps and parts. Give reasons for each step.

(b) Finding the CDF $$F_X(t):$$ Let $$0

$$F_X(t) = P(X \le t) = \int_{-\infty}^0 0\, ds + \int_0^t \frac 1 k\, dk = \cdots = t/k.$$ What are values of $$F_X(t)$$ for $$t < 0$$ and for $$t > k?$$ You will already have found $$E(X_1 + X_2).$$ Also find $$Var(X_1+X_2);$$ you will need it later.

(d) Let $$M = \max(X_1, X_2).$$ What is the CDF of $$M?$$ Again start with $$0 < t < k.$$

$$F_M(t) = P(M \le t) = P(X_1 \le t,\, X_2 \le t)\\ = P(X_i \le t)P(X_2 \le t) = (t/k)^2.$$ Now take the derivative to find the (non-uniform) density $$f_M(t)$$ of $$M$$ and use it to find $$E(M)$$ and $$Var(M).$$

(f) Compare the variances of the two unbiased estimators, based on variances found above.

I used R statistical software to simulate many runs of this two-observation experiments with $$k = 4,$$ and thus to make histograms suggesting the distributions of $$S=X_1+X_2$$ and $$M^\prime = 1.5M.$$ You can see from the histograms which of these unbiased estimators has the smaller variance and hence is the preferred estimator. (This is just for intuition; you are not expected to show a simulation as part of your answer.) 