Statistics: probability distribution problem

In summary: I have not seen any answer for (b) or (c).In summary, we have two independent random variables, ##X_1## and ##X_2##, with a uniform distribution on the interval [0,k]. The cumulative distribution function for each variable is ##F_{X_i}(x) = \frac{x}{k}## for x ≤ k and 0 otherwise. For part (b), we need to find the probability distribution of M, the larger of the two variables. This can be done by writing the CDF of the sum as a double integral, with the limits for the inner integral dependent on the outer integration variable. For part (c), we need to show that 1.5M is
  • #1
chwala
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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##
 
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  • #2
chwala said:
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.
 
  • #3
chwala said:

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)?
 
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  • #4
Ray Vickson said:
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..
 
  • #5
chwala said:
That is how it is exactly on the textbook..
I asked you two questions, and you have not answered either of them.
 
  • #6
Ray Vickson said:
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##
 
  • #7
Ray Vickson said:
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...
 
  • #8
andrewkirk said:
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...
 
  • #9
andrewkirk said:
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##
 
  • #10
Any feedback?
 
  • #11
chwala said:
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).
 
  • #12
chwala said:
Any feedback?

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

1. What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of different outcomes of a random event. It assigns probabilities to all possible outcomes, with the total probability equaling 1.

2. How is a probability distribution different from a probability?

A probability distribution is a function that gives the probabilities of all possible outcomes, whereas a probability is a single value that represents the likelihood of a specific outcome occurring.

3. What is the relationship between probability distributions and statistics?

Probability distributions are a fundamental concept in statistics, as they allow us to model and analyze random events. They are used to calculate probabilities and make predictions based on data.

4. What is the difference between discrete and continuous probability distributions?

A discrete probability distribution is one where the outcomes are distinct and countable, while a continuous probability distribution is one where the outcomes are infinite and uncountable. Discrete distributions are typically used for discrete data, while continuous distributions are used for continuous data.

5. How can probability distributions be used in real-life situations?

Probability distributions are used in a wide range of fields, including finance, engineering, and medicine, to make predictions and inform decision making. They can be used to model and analyze data, assess risks, and make forecasts about future events.

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