Dtjytyk

How does the math work for Perception checks?

What exact color does ozone gas have?

Does the Linux kernel need a file system to run?

Mimic lecturing on blackboard, facing audience

Why "had" in "[something] we would have made had we used [something]"?

Does Doodling or Improvising on the Piano Have Any Benefits?

What should you do when eye contact makes your subordinate uncomfortable?

What is going on with 'gets(stdin)' on the site coderbyte?

Non-trope happy ending?

Does IPv6 have similar concept of network mask?

What are the advantages of simplicial model categories over non-simplicial ones?

Has any country ever had 2 former presidents in jail simultaneously?

Are Captain Marvel's powers affected by Thanos' actions in Infinity War

Terse Method to Swap Lowest for Highest?

What are some good ways to treat frozen vegetables such that they behave like fresh vegetables when stir frying them?

How does a computer interpret real numbers?

On a tidally locked planet, would time be quantized?

Do the primes contain an infinite almost arithmetic progression?

What's the difference between releasing hormones and tropic hormones?

Does malloc reserve more space while allocating memory?

The IT department bottlenecks progress. How should I handle this?

Why does the Sun have different day lengths, but not the gas giants?

Why is so much work done on numerical verification of the Riemann Hypothesis?

Open a doc from terminal, but not by its name

How to construct an independent sequence of random variables

Construction of independent random variablesSum of two independent exponential distributionsTesting for the independence of random variablesQuestion about sum of chi-squared distributionConstruct a sequence of i.i.d random variables with a given a distribution functionProbability Theory - Transformation of independent continuous random variablesTransformation of random variables exerciseProve a central limit theorem for $T_{n}=sum^{n}_{i=1}Y_i$. For r.v's 1-dependent.Sequences of exponential random variablesHow to show that a particular distribution is the Lebesgue measure

0

$begingroup$

Let $Q$ be a probability measure on $mathbb{R}$ with distribution function $F$. The inverse distribution function $Y$ on $(0,1)$ is defined by:
$$Y(w) = sup,{xmid F(x)<w}, quad0leq wleq 1$$

(i) Show that the distribution of $Y$ is $Q$.
(ii) For $Q_1,Q_2,..$ a sequence of prob measures on $mathbb{R}$, construct a sequence of r.v's $Y_1,Y_2,..$ on $((0,1),mathcal{B},lambda)$ that are independent and the distribution of $Y_i$ is $Q_i$.

My attempt: $F_Y (y) = mathbb{P}(Yleq y) = lambda{w: Y(w)leq y} = lambda{w: wleq F(y)} = F(y)$ and hence the distribution of $Y$ is $Q$. For (ii), I can construct a sequence of inverse distribution functions $Y_i$ such that the distribution of $Y_i$ is $Q_i$ but how can I show that they are independent? Thanks and appreciate a hint.

real-analysis probability-theory statistics

share|cite|improve this question

edited Mar 13 at 16:58

Brian

892116

asked Mar 13 at 16:47

manifoldedmanifolded

49019

$endgroup$

add a comment | 

0

$begingroup$

Let $Q$ be a probability measure on $mathbb{R}$ with distribution function $F$. The inverse distribution function $Y$ on $(0,1)$ is defined by:
$$Y(w) = sup,{xmid F(x)<w}, quad0leq wleq 1$$

(i) Show that the distribution of $Y$ is $Q$.
(ii) For $Q_1,Q_2,..$ a sequence of prob measures on $mathbb{R}$, construct a sequence of r.v's $Y_1,Y_2,..$ on $((0,1),mathcal{B},lambda)$ that are independent and the distribution of $Y_i$ is $Q_i$.

My attempt: $F_Y (y) = mathbb{P}(Yleq y) = lambda{w: Y(w)leq y} = lambda{w: wleq F(y)} = F(y)$ and hence the distribution of $Y$ is $Q$. For (ii), I can construct a sequence of inverse distribution functions $Y_i$ such that the distribution of $Y_i$ is $Q_i$ but how can I show that they are independent? Thanks and appreciate a hint.

real-analysis probability-theory statistics

share|cite|improve this question

edited Mar 13 at 16:58

Brian

892116

asked Mar 13 at 16:47

manifoldedmanifolded

49019

$endgroup$

add a comment | 

$begingroup$

Let $Q$ be a probability measure on $mathbb{R}$ with distribution function $F$. The inverse distribution function $Y$ on $(0,1)$ is defined by:
$$Y(w) = sup,{xmid F(x)<w}, quad0leq wleq 1$$

(i) Show that the distribution of $Y$ is $Q$.
(ii) For $Q_1,Q_2,..$ a sequence of prob measures on $mathbb{R}$, construct a sequence of r.v's $Y_1,Y_2,..$ on $((0,1),mathcal{B},lambda)$ that are independent and the distribution of $Y_i$ is $Q_i$.

My attempt: $F_Y (y) = mathbb{P}(Yleq y) = lambda{w: Y(w)leq y} = lambda{w: wleq F(y)} = F(y)$ and hence the distribution of $Y$ is $Q$. For (ii), I can construct a sequence of inverse distribution functions $Y_i$ such that the distribution of $Y_i$ is $Q_i$ but how can I show that they are independent? Thanks and appreciate a hint.

real-analysis probability-theory statistics

share|cite|improve this question

edited Mar 13 at 16:58

Brian

892116

asked Mar 13 at 16:47

manifoldedmanifolded

49019

$endgroup$

share|cite|improve this question

edited Mar 13 at 16:58

Brian

892116

asked Mar 13 at 16:47

manifoldedmanifolded

49019

share|cite|improve this question

edited Mar 13 at 16:58

Brian

892116

asked Mar 13 at 16:47

manifoldedmanifolded

49019

edited Mar 13 at 16:58

Brian

892116

edited Mar 13 at 16:58

Brian

892116

asked Mar 13 at 16:47

manifoldedmanifolded

49019

asked Mar 13 at 16:47

manifoldedmanifolded

49019