Statistics: probability distribution problem

[chwala]

Homework Statement

two indepedent observations $X_1$ and $X_2$ are made up of the continuous random variable having the probability density function $f(x)= 1/k$, and $0≤x≤k$
find a. the cumulative distribution of $X$
b. Find the probability distribution of M, the larger of $X_1$ and $X_2$
c. Show that 1.5M is an unbiased estimator of$k$.

Homework Equations

The Attempt at a Solution

part (a)
given $f(x)= 1/k,$, then cdf, $F(X)=∫ {1/k}dx$ = $1$
limits from $0$ to $k$
$k=1$→ the cumulative distribution function , $F(X)=x$

 

[andrewkirk]

two indepedent observations $X_1$ and $X_2$ are made up of the continuous random variable

This is an incorrect and confusing way to say it. Only one observation can be made of any random variable. So what we have is that there are two random variables $X_1$ and $X_2$ that are independent and each of which has the distribution you describe - a uniform distribution on $[0,k]$.

If $F_{X_i}$ is the cumulative distribution function for $X_1$ and $X_2$ and $f_{X_i}$ is the probability density function, then
$$F_{X_i}(x)=\int_0^x f_{X_i}(u)\,du$$
What is the result of that integral, given that we've been told that $f_{X_i}(u)=\frac1k$ for $u\in[0,k]$?
It is not 1 unless $x=k$.

For (b) you will need to write the CDF of the sum as a double integral, over possible values of $X_1$ and $X_2$. The limits for the inner integral will depend on the value of the outer integration variable.

 

[Ray Vickson]

Homework Statement

two indepedent observations $X_1$ and $X_2$ are made up of the continuous random variable having the probability density function $f(x)= 1/k$, and $0≤x≤k$
find a. the cumulative distribution of $X$
b. Find the probability distribution of M, the larger of $X_1$ and $X_2$
c. Show that 1.5M is an unbiased estimator of$k$.

Homework Equations

The Attempt at a Solution

part (a)
given $f(x)= 1/k,$, then cdf, $F(X)=∫ {1/k}dx$ = $1$
limits from $0$ to $k$
$k=1$→ the cumulative distribution function , $F(X)=x$

Your question (a) is unclear. Does it ask for the distribution of $X_1+X_2?$

What is your approach to (b)?

 

[chwala]

Your question (a) is unclear. Does it ask for the distribution of $X_1+X_2?$

What is your approach to (b)?

That is how it is exactly on the textbook..

 

[Ray Vickson]

That is how it is exactly on the textbook..

I asked you two questions, and you have not answered either of them.

 

[chwala]

I asked you two questions, and you have not answered either of them.

sorry was a bit held up...i am not sure, i am assuming that the the distribution applies to both $x_1$ and $x_2$

 

[chwala]

I asked you two questions, and you have not answered either of them.

see post two above, only one observation can be made for a random variable, hope that answers that...

 

[chwala]

This is an incorrect and confusing way to say it. Only one observation can be made of any random variable. So what we have is that there are two random variables $X_1$ and $X_2$ that are independent and each of which has the distribution you describe - a uniform distribution on $[0,k]$.

If $F_{X_i}$ is the cumulative distribution function for $X_1$ and $X_2$ and $f_{X_i}$ is the probability density function, then
$$F_{X_i}(x)=\int_0^x f_{X_i}(u)\,du$$
What is the result of that integral, given that we've been told that $f_{X_i}(u)=\frac1k$ for $u\in[0,k]$?
It is not 1 unless $x=k$.

For (b) you will need to write the CDF of the sum as a double integral, over possible values of $X_1$ and $X_2$. The limits for the inner integral will depend on the value of the outer integration variable.

for part (a),
$\int_0^k k^-1\,dx$ = $x/k$ using given limits and substituting for $x$, we get $1-0 = 1$...unless i am missing something here...

 

[chwala]

This is an incorrect and confusing way to say it. Only one observation can be made of any random variable. So what we have is that there are two random variables $X_1$ and $X_2$ that are independent and each of which has the distribution you describe - a uniform distribution on $[0,k]$.

If $F_{X_i}$ is the cumulative distribution function for $X_1$ and $X_2$ and $f_{X_i}$ is the probability density function, then
$$F_{X_i}(x)=\int_0^x f_{X_i}(u)\,du$$
What is the result of that integral, given that we've been told that $f_{X_i}(u)=\frac1k$ for $u\in[0,k]$?
It is not 1 unless $x=k$.

For (b) you will need to write the CDF of the sum as a double integral, over possible values of $X_1$ and $X_2$. The limits for the inner integral will depend on the value of the outer integration variable.

therfore for part (a),
$x≤0, F(x)=0$
$0≤x≤k, F(x)= x/k$
$x≥k, F(x) = 1$

 

[chwala]

Any feedback?

 

[andrewkirk]

therfore for part (a),
$x≤0, F(x)=0$
$0≤x≤k, F(x)= x/k$
$x≥k, F(x) = 1$

Replace $F$ by $F_{X_i}$, meaning the cumulative distribution function of random variable $X_i$ (for $i\in\{1,2\}$) and what you have written will be correct for (a).

 

[Ray Vickson]

Any feedback?

Yes. What is your answer to (b)? Ditto for (c).