Convergence of random variables Puns

Suppose that we have (to = arrow that means converges to, I think there is an encoding error) X_n to X and Y_n to Y both almost surely (a.s) or in density (d).

I found online that X_n + Y_n to X+ Y and X_n * Y_n to X*Y with the same thing we suppose as our hypothesis (a.s or d).

But I could not find if we could generalise at this : (X_n , Y_n ) to (X,Y) as a couple of variable so we can use the continuous mapping theorem to generalise the fonction to more complex ones than just add and multiply ?

So can we say that the couple (X_n , Y_n ) converge (a.s) or (d) if X_n and Y_n converge separately (a.s) or (d) ?

I have asked in another sub but think this one would be more suited.

And after some though I think I have an answer (see below) but I am not sure at all, so if somebody has the answer let me know ! :)

I think that we can do this but I am not sure :

For (d) : Let a>0 We look at P( || (Xn,Yn) - (X,Y) || ) inferior or = P( || Xn-X || ) + P( || Yn - Y || ) to 0 because Yn and Xn converge (d) to Y and X

For (a.s): We have w in universe such (Xn,Yn)(w) = (Xn(w),Yn(w) ) And since for all w except a subset of measure null by P (or anything that dominates P) we have (Xn(w),Yn(w) -> (X(w),Y(w) ) = (X,Y)(w) because Xn and Yn converge both to X and Y (a.s) So (Xn,Yn) to (X,Y) (a.s)

Are my justification right ? I am not sure at all but tried to be as precise as I could .

Thanks in advance

https://imgur.com/C8JechQ

It's a true or false question. Pretty sure that this is false, but I'm struggling to come up with a counterexample for this.

So I've run across a problem that I have to know these topics by tomorrow. Could anyone give me a run down of these topics?

Here are three sample questions (practice questions in a study guide) for each one that I would love for someone to answer. These are not homework and maybe I should have posted on /r/learnmath, but I decided to post here.

Condition Distributions:

1. Define the conditional variance of Y given X. show that var(Y) = E(var(Y|X)) + var(E(Y|X)).

2. Let X,Y be iid uniform [0,1], and Z = X+Y. What is E(E(X|Z))? What is var(E(X|Z))?

3. Let X,Y be independent and have Pois(2) distributions. What is the distribution of X given X+Y=4?

Convergence of Random Variables

1. Let X_n ~ Bin(n,lambda/n) and Y ~ Pois(lambda). Show that X_n -> Y. (The arrow has a D on top of it.)

2. Show that with probability 1, a random walk with 2/3 bias (i.e. a +1 step with problem 2/3, -1 with prob 1/3, hits 0 only finitely man times.

3. Let {X_n} be a sequence of indepedent random variables where X_n = 1 with probability 1/n and 0 with probability 1 - 1/n. Show that X_n does not converge to 0 a.s.

Limit Theorems

1. Let X_i's be iid with finite mean and variance. What can you say about (*) as n-> infinity?

2. Let Y_n~Pois(n). find the proper scaling so that Y_n converges to a N(0,1) and prove that it converges.

3. Prove a Chernoff bound for the sum of iid Uniform [0,1] random variables.

(*) = Summation from i to n (X_i - EX_i)/n^(2/3)

1. If we let X1, X2, X3, X4, X5 be independent continuous random variables with a common distribution function F and density function f, what is the probability X3 < X1 < X4?

2. Let X1, X2, ... Xn be independent identically distributed random variables with density f(x)=e^-x from 0 to infinity. For which values of a>0 does P[Xn > log n + a log log n i.o.] =1? I.O. = infinitely often. I know this involves the Borel-Cantelli lemma, but I don't know what to do.

3. For a nonnegative continous random variable W with distribution function F, show that E[W^2 ] = integral (0 to infinity) 2y(1-F(y))dy

4. Suppose that 80 balls are placed into 40 boxes at random and independently. What is the variance of the number of empty boxes?

I know how to get the expected value of empty boxes using indicator random variables, but how would I go about finding the variance? I realize Var(X)=E(X^2 ) - (E(X))^2 , but how would I find E(X^2 )? Or is there another way of finding the variance?

Convergence Problems

5. Let X1, X2, ... be independent random variables with Uniform (0,1) distribution. For each integer n > 1 let Mn denote the minimum of (X1, X2, ..., Xn). Show that nMn converges in distribution, and identify the limiting distribution.

6. Let X1, X2, X3, ... be independent identically-distributed random variables with a Poisson(λ) distribution. What is the probability Xn > n infinitely many times? Is your answer different if the random variables are not dependent?

7. Let X1, X2, ... be a sequence of independent identically distributed Poisson random variables with mean 5. Does (X1^2 + X2^2 + ... + Xn^2) / n converge in probability? If so, what's the limit? If not, explain why. Does it converge almost surely? If so, what is the limit? If not, explain why?

8. Let X1, X2, ... be a sequence of nonnegative independent random variables, all with the same distribution given by density function f(x)=4x(1-x) from 0 to 1. Describe the limiting behavior (X1 + X2 + X3 + ... + Xn)/n as n approaches infinity. Which limit theorems apply, and which modes of convergence are involved?

I've tried reading the theory behind the convergence of random variables, but I've found it hard to understand how to apply them to problems.

Help with any of these problems is appreciated. Thank you.

So I have a homework asking me to find a random variable converging to probability. I have no idea you can find a new random variable excpet copying the pdf of the distribution provided (in my case it is gamma distribution). Can I ask what I should do to find a random variable that would converge to a certain probability? Given that the probability is only in the form of f(m|a,b).

Let P be the transition matrix and n the number of random walks. If you rearrange the matrix such that the transient states (those who are eventually left forever) come first and the absorbing states (those who transition back to themselves with probability 1 each time step), you can partition P into

[Q R] [0 I ]

where Q is the transition probabilities between transient states and R is the probability of transitioning to each absorbing state from each transient state. The fundamental matrix N gives the expected throughput of each transient state and is calculated as N=[I-Q]^-1 where I is an identity matrix of equal dimension to Q. The formula is the matrix version of the geometric series, as it comes from N=I+Q+Q^2+Q^3… up to infinity. You can then calculate the absorption probabilities A from A=NR, giving the probability of absorption into each absorbing state given the initial state.

If the elements of Q are uniform random variables and you take n random walks on this markov chain, what are the distributions of the elements of N and A? Assume that the elements of Q are independently but not necessarily identically distributed.

Given means and variances of a finite set of random variables find variance of their linear combination.

First I need to scale each variable, and it means just multiplying the mean and the variance by corresponding scalar.

Then mean of a sum is a sum of means.

Can you explain to me how to find covariance?

I try case with two RVs and variance should be var_1+var_2 +2*covariance, right?

I get an answer order of magnitude off expected value. What might I do wrong?

I have a requirement to generate a seeded Random number server-side in ASP.NET Core inside a controller so that the generated random value can always be replicated based on the seed value and iteration N.

At the moment I'm recreating Random each time and iteration from 0...N to get the next iteration but this gets progressively slower the more iterations there are. Eg something like this

public int NextIteration(int seed, int iteration)
{
    Random rand = new Random(seed);
    int r = 0;
    for (int n = 1; n <= iteration; n++)
    {
        r = rand.Next();
    }
    
    return r;
}

Is there a way to store complex objects as session variable without destroying it's state?

Random rand = new Random(seed)

I have looked at session variables but this requires serializing/de-serializing which i) loses the state of the Random generator anyway and ii) is often slower than creating a new instance and iterating

Is there another way to preserve state of the Random generator that doesn't require serializing/de-serializing and is user specific?

So my project is going to have a lot of classes of various types of 'pets', and I need a quick way of being able to return, randomly, one of these pets, preferably without a switch statement that is 100 lines long.

For example I'd want it to behave something like this:

Pet classes: Dog, Chicken, Ant, Tiger, Crocodile, Fish, etc...

Code:

var rndPet = getRandomPet();

function getRandomPet() {

return random(<pet classes>);

}

All I can think of right now is something basic like:

function getRandomPet() {

switch (pets) {

case "ant": return new Ant();

break;

case "dog": return new Dog();

break;

...

}

}

I’m about to calculate the expectation of Z = X(t)exp(X(t)) where X(t) is a brownish motion with drift.

But how do I calculate an expectation of a product of 2 random variables if they are not independent?

I am at my absolute wits end with this one. I am attempting to find the pdf of the sum of two random variables using the convolution method, however, I cannot find the limits of the integral. I have spent hours trying to figure this out but there's very little documentation on the internet.

Let X be a uniform continuous random variable on (0,2), and Y be a continuous uniform variable on (0,3). Find the pdf of Z = X+Y by method of convolution. Can anyone give me an in-depth way of finding the limits of the integral and possibly a general method for two arbitary random variables?

I’m guessing this is not illegal, but the wages are pissing people off and I think it’s for good reason. Wondering if I can reason with my boss being part of management

Starting salary for probationary period is $10. After that, increase to $11. Up until this year, the only way to get a raise was through work-related “projects”. Not merit based. A few of the people in my team benefited from that, but no one except for the most senior employee ever did (or were encouraged to do) more than two projects.

Starting in the summer, they made two low-level managerial positions that bumped pay up by $2.

Because we had a bunch of people leave us (for low pay and environment) boss raised wages $1 for <30 hours a week, and $2 for 30+ hours.

This means that there is an employee who has worked here for 2 years, that only was given one project opportunity, is making the same $13 as our newest hire, and $2 less than a coworker who worked just as long as her. I believe her being ADHD and not quite as “normal” of a worker has affected her chances quite a bit, though without good cause.

Additionally - an ex coworker just confided in me that the bosses started her at $13 for probationary and $14 after the bump up to “help her out” with housing since she went to the same college as the boss - when they have literally turned down candidates with actual field experience who wanted that same $13/$14 initial pay rate.

Is there anything to be done about this? Is any of this against labor laws (the college shoe-in making more than any other employee for the same job title, for instance)?

Right now I’m trying to help the senior employee making the same as new hires put together a solid raise request plan and get a few positive recommendations from coworkers and clients to boost that, but it feels like that’s all I can do.

Also f Reddit app for making this take 5 times as long to type out.

I'm honestly not even sure how to Google this. We can do things like adding random variables, so there's distribution families which are closed under addition (eg binomial for fixed p is closed under addition). Is there a notion of a basis random variables for a family (eg Bernoulli(p) is enough to generate binomial (n,p), but sum(ai xi) for xi iid Bernoulli spans much more-- what I'm not sure).

Is there an area of probability where questions like these fit? An abstract algebra of random variables?

Just a quick question for any statisticians out there. Im solving a homework question and I got to the expression Cov(X,XY).

Is it mathematically correct to say that Co(X,XY) equals Cov(X,X) * Cov(X,Y) and therefore equals Cov(X,Y)?

Thanks

I use twine sugarcube v2.3.14.

I have different locations (buildings) in my game the player can control. Those can be attacked by the enemy. How many buildings and what buildings the player controls can vary.

Now i want the enemy to attack a random building out of the players buildings. Therefore i have to check which buildings are under the players control and then chose a random one. I check if a player controls a building like that:

<<if $bHQ gt 0 and $bHQ lte 10>>
code
<</if>>
<<if $bBarracks gt 0 and $bBarracks lte 10>>
code
<</if>>
and so forth ...

I had a similiar problem before, what i did was this:

<<if _tools is true and _surprise is false and _other is false>>
	<<set _rollEventVillage = "tools">>
<<elseif _tools is false and _surprise is true and _other is false>>
	<<set _rollEventVillage = "surprise">>
<<elseif _tools is false and _surprise is false and _other is true>>
	<<set _rollEventVillage = "other">>
<<elseif _tools is true and _surprise is true and _other is false>>
	<<set _rollEventVillage = either("tools","surprise")>>
<<elseif _tools is true and _surprise is false and _other is true>>
	<<set _rollEventVillage = either("tools","other")>>
<<elseif _tools is false and _surprise is true and _other is true>>
	<<set _rollEventVillage = either("surprise","other")>>
<<elseif _tools is true and _surprise is true and _other is true>>
	<<set _rollEventVillage = either("tools","surprise","other")>>
<<elseif _tools is false and _surprise is false and _other is false>>
	<<set _rollEventVillage = "none">>
<</if>>

But that were only 3 variables, now i have about 15 different variables, if i would do the same, the length of the code would be crazy. I thought i could do something like this:

<<set _numberBuildings = 0>>
<<set $target1 = "none">>
<<set $target2 = "none">>
<<set $target3 = "none">>
and so on ...

<<if $bHQ gt 0 and $bHQ lte 10>>
    <<set _numberBuildings += 1>>
    <<set $target1 = "bHQ">>
<</if>>
<<if $bBarracks gt 0 and $bBarracks lte 10>&gt

Basically what the title says.

I hate and love it at the same time.
Please don't ask me how this works or to add a feature.

Github Repo

namespace RandomMemberNameGenerator
{
    class Program
    {
        enum xTmpRandom
        {
            STRUCTUREBOOLEAN,
            ifTmpArray,
            BaseJFrom,
            linq_dns,
            adddict
        }

        static List<string> GlobalizationDefault;
        static Random PUTSHARED = new Random();
        static TextInfo SYNCOUTPRIVATE;

        static void Main()
        {
            GlobalizationDefault = File.ReadAllText(@".\words.txt").Split(Environment.NewLine).ToList();
            SYNCOUTPRIVATE = CultureInfo.CurrentCulture.TextInfo;

            Console.WriteLine("Press a key to generate a name");

            while (async_delegate_dictionary())
            {
                Console.ReadKey();
                string ofByte = string.Empty;
                Console.WriteLine(firstFinallyFirst(ITERATIONPOSTENUM(ofByte, 0, PUTSHARED.Next(2, 4))).Replace(" ", ""));
            }

        }

        private static string ITERATIONPOSTENUM(string serializestructurefrom, int ASYNCPARTSELECT, int virtualdatetime_)
        {
            string defaultZipDatetime = GlobalizationDefault[PUTSHARED.Next(0, GlobalizationDefault.Count)];
            if (int.TryParse(defaultZipDatetime, out _))
                return ITERATIONPOSTENUM(serializestructurefrom, ASYNCPARTSELECT, virtualdatetime_);

            serializestructurefrom += defaultZipDatetime + " ";

            if (ASYNCPARTSELECT == virtualdatetime_) return serializestructurefrom;

            ASYNCPARTSELECT++;
            return ITERATIONPOSTENUM(serializestructurefrom, ASYNCPARTSELECT, virtualdatetime_);
        }

        private static string firstFinallyFirst(string func_integer_date_png)
        {
            return (PUTSHARED.Next(0, Enum.GetValues(typeof(xTmpRandom)).Length)) switch
            {
                (int)xTmpRandom.STRUCTUREBOOLEAN => readCatchMain(func_integer_date_png),
                (int)xTmpRandom.ifTmpArray => mainchrinteger(func_integer_date_png),

Just wondering how I could do this without a chunk of code for each variable.

I’ve spent so long trying to figure out what the name is. If it even exists. Basically trying to calculate how many trials required for x amount of successes/failures. For example you have a given P, and want to know the average amount of trials until (10,20,1,000) successes.

Geometric random variable is the closest I’ve found, but it stops after a single success. With the other being negative binomial distribution, but needs a given amount of trials. Am I just overlooking something and making this harder than it should be? Or is there a specific name for this?

Edit: found a website with a calculator for what I’m looking for, not entirely sure what the equation is supposed to equal... https://www.anesi.com/negativebinomial.htm?p=0.5&r=20

I’ts been a while since I’ve done statistics and I came about a problem where there’s two i.i.d. discrete uniform random variables both from [1,20] and I have to find the expected value of the maximum between the two. I’m not really sure how order statistics is supposed to work for discrete distributions. Any thoughts?

I need to show that for a random variable X, and Y = max(c, X), Var(Y) <= Var(X). c is a constant.

My thought is that considering values in R < c, Y has a point mass at c and the variance of Y would be 0, versus whatever the variance of X is, which would nonnegative. Then for values in R greater than or equal to c, the variance of Y would be the same as the variance of X.

I feel a little weird about splitting up the variance like this (would the law of total variance apply?) and don’t know if I’m even thinking about this the right way. If I am, how can I make what I’m thinking more rigorous? If I’m not, can someone provide some intuition?

Thanks

Edited

Addendum I’m back with some more thoughts. I think this should be split into 2 cases: c is in the support of X or c isn’t in the support of X. I think the first case reduces to varY = 0 if c is greater than the max value of the support since P(Y=c) = 1 or varY = varX if c is less than the minimum value in the support of X, since P(X > c) = 1.

So that leaves the second case where c is in the support of X, (I’m going to write this as S(X)). In this case, EX <= EY as a commenter said. However, I’m not sure how to use this. Expanding varY = E(Y - EY)^2 = sum over x < c in S(X): p(x)(c-EY)^2 + sum over x >= c in S(X): p(x)(x-EY)^2

I’m not sure where to go from here.

SO I'm trying to explore the statistical properties of including vs. excluding a particular predictor 'A' in a classification context, trying to classify some binary 'B', specifically to probe the inverse question -- whether the effects of that feature 'B' have been removed appropriately from the predictor A by a colleague's procedure.

My thinking was that if the effect of 'B' still echoes through the "B-corrected" data 'A', we should be able to retrieve information about 'B' from the data using 'A', along with the rest of our data.

Of course, adding variables will always improve retrodictive classification accuracy. I envision two easy ways around this -- cross-validation, and permutation of the predictor 'A'.

I decided to start with the latter: basically, evaluating classification accuracy of 'B' from my data (incl. 'A'), and then computing a "null distribution" of classification accuracy by permuting (i.e. resampling with replacement) 'A' some number of times and re-running the classification algorithm each time.

So in the end, I might get a retrodictive classification accuracy using my real data of X = 75 / 100, and permuted accuracies of Y = (74, 71, 74, 73, 76, 75, 75, 76, 74, ...) / 100. My thinking is that if my colleague adequately removed all trace of the feature from the feature in question, the quantile of my real accuracy should be uniform(0,1) within my permuted accuracy (colleague actually ran this "correction" independently on thousands of features, so I could test for uniformity here).

Question is, I'm not really sure how to find the quantile of my discrete random variable, i.e. the count of correct classifications out of 100. The usual way I'd do this would be to do something like n_(X > Y) / n_Y, so if you truncate Y above before the ellipse, you get 5/9 = 0.56-ish. Or should I do n_(X >= Y) / n_Y, in which case I have 7 / 9 = 0.78-ish? In the continuous case it doesn't matter because we never get exact equality, but here it does. CDFs of ordered discrete random variables include the # being evaluated while summing over their PMFs, but IDK if that really provides principled guidance.

I can also imagine calculating the quantile after removing all the proportions in Y that match X exactly. This seems intuitive to me -- e.g. imagine we only have 3 observations, (1,2,3), and our distribution of permuted accuracies contains 50 '1/3's, 100 '2/3's, and 50 '3/3's. Using the real data, we observe a 2/3 -- seems nicer for it to fa

Suppose that we have X_n -> X and Y_n -> Y both almost surely (a.s) or in density (d).

I found online that X_n + Y_n -> X + Y and X_n * Y_n -> X*Y with the same thing we suppose as our hypothesis (a.s or d).

But I could not find if we could generalise at this : (X_n , Y_n ) -> (X,Y) as a couple of variable so we can use the continuous mapping theorem to generalise the fonction to more complex ones than just add and multiply ?

So can we say that the couple (X_n , Y_n ) converge (a.s) or (d) if X_n and Y_n converge separately (a.s) or (d) ?