Chi squared distribution Puns

[Q] What is the distribution that can result from squaring the Chi-Squared distribution? i.e. I have a Chi-Squared distributed random variable and would like to apply hypothesis tests on its power.

I get that it works, but why?

For the probability distributions I've studied up to this point, I never had to ask this question because it was always obvious.

Take the normal distribution. Say my battery dies after a short period of just x months. The manufacturer says batter life of all batteries in this range is normally distributed with a mean of μ months and a standard deviation of σ months. I know the normal distribution and I know how to use that knowledge to check likelihood of my battery by consulting a z table. The connection is intrinsic and it makes sense.

Similarly, I know the binomial distribution and I can easily see that it's my go to distribution for working out the probability of a specific outcome of multiple coin tosses. It's also an intrinsic connection.

Now, back to chi-square. My first issue is with the abstract nature chi-square distribution itself. I struggle to see how it can be tangibly related with real world data.

In trying to understand this distribution, I manually built a chi-square distributed dataset in R to test if I understood the mechanics of the distribution. I understood the mechanics, but that didn't help me understand how it related to the real world in the same way that the normal or binomial distributions do.

Also, I can clearly see that a chi-square goodness of fit test resolves a problem between expected and observed distributions, but I don't get why the chi-square distribution reveals this. They seem like two separate concepts and I'm struggling to see the relationship.

Here's what I understand to be a comparison of the chi-square dataset I built and a goodness of fit test.

In the chi-square dataset I built, I squared k number of randomly sampled values from a normal distribution and added them together. My degrees of freedom was equal to k.

The goodness of fit test involves finding the error between observations for k number of bins (e.g. days) with the expected value for each bin. To do this, for each bin (K1-Ki), I subtract expected value of K from the observed value of K, square that and divide by the expected value of K. I then add all these together. My degrees of freedom are K-1.

Here are the similarities I do see. We have a value for k in both scenarios. We also do some squaring for each value of k.

Here are the differences I understand. We're subtracting expected from observed because we're interested in the distribution of the error.

Here are the differences I don't understand. Why are we

I came across some statistics a coworker performed at work and I need some help understanding chi squared distributions.  Some of my numbers and wording are a bit vague so bear with me.

Let's say that I am given a data set with 10 values.  The mean is 0.7 and the standard deviation is 1.27.  this is the observed data set.

We want to find the best and worst case scenarios for the next ten years with 95% confidence this is a Mean Time to Failure problem. The person who did the work said they achieved this by a "chi square distribution used to determine upper and lower 95% confidence on best/worst case data."

They used the mean of 0.7 for their expected best case scenario.  No standard deviation is given.  I actually think their data set is composed of the value 0.7 ten times in a row but I could be wrong.

They use a mean of 1.4 for the expected worst case scenario with no standard deviation given.  I believe the data set is the value 1.4 ten times in a row.  This value came from looking at the same population as the observed data set but from many years previous.

We end up with a best case scenario of 0.2 (mean value) and a worst case scenario of 2.3 (mean value) (Upper and Lower Bounds for the confidence interval at 95% confidence).

I am having a hard time figuring out how my coworker calculated this confidence interval because all of the examples I have found online determine the confidence interval using standard deviations and not the mean. Can anybody help me out?  Is it even possible to calculate a Chi Squared Distribution and Confidence interval using means? I have looked in several places online and am still confused.

Can someone please direct me to where I can look at the proof of this?

sum(Xi - sample mean) ~ (sigma^2 ) chi-squared(n-1)

I’m trying to look it up but I don’t know what to even put into google to get the correct results. (Could you also tell me what you googled if possible?)

I'm studying physics at university and I can't quite get the real meaning of this distribution. Probably it has something to do with the Euler's formula, I don't know what is its meaning too. Lastly: how can I determine if a variable is a chi-squared variable?

I am doing some independent study on introductory Black-Body radiation. I took an AP Statistics class my senior year in High School (Just graduated!). I loved the class and I know how relevant statistics is in Physics. My question arose due to my studies about how the graphs of black-body radiators are mentioned to have probabilities of various frequencies. If I am on the right thought process (or wrong) any help would be splendid. So if anyone could provide an insight to my question about whether the distribution of a black-body radiatior graph is a Chi-Squared distribution that would be wonderful. Thank you for your time.

Hello again. I'm stuck on two sort of similar questions...

First question: I have a discrete random variable Y such that P(Y=1) = P(Y=-1) = 1/2. I also have Z~N(0,1) and it is given that Y and Z are independent. I must show that X = YZ also has the standard normal distribution. I must also show that X and Y are uncorrelated.

Second question: I have X and Y, iid both ~N(0,1). I need to find the joint mgf of U and V, where U = X + Y and V = X^2 + Y^2 .

Ok, so here's what I got for each one so far:

First question, I have to do two things, (1) show that X~N(0,1) and (2) show that X and Y are uncorrelated. So far, what I have found is that independence implies uncorrelation, (but not necessarily vice versa). [This is due to the fact that independence between random variables A and B means E(AB) = E(A)E(B), whereas the definition of covariance is E(AB) - E(A)E(B), thus making covariance, and so correlation, zero.] Thus, if I show E(XY) = E(X)E(Y), I’m good for the second part of this question. So let’s assume I have already shown that X~N(0,1), meaning the expectation of X is 0. Using the values given for Y, we can see that Y also has an expectation of 0. We were told Y and Z are independent, so we know E(YZ) = E(Y)E(Z). We just showed E(Y) = E(Z) = 0, so E(YZ) = 0. Since X is defined as YZ, we know E(X) = 0. Therefore, given that E(X) and E(Y) are 0, we know that it is true E(XY) = E(X)E(Y) and thus they are independent. But is this accurate?? I feel like it’s sort of ‘cheating’ using this equation for independence since all the expectations are equal to zero. As for the first part of this question, I’m sort of lost. What is the best method to try and show the distribution of a random variable, especially one that is defined to be the product of two others? I’m especially confused because Y is discrete while Z is continuous.

For the second question, I know that U = X + Y ~ N(mu_x + mu_y , sigma^2 _x + sigma^2 _y). And so U ~ N(0,2). Also, by the definition of a chi squared random variable, V is a chi squared of degree 2. I also know the joint pdf of X and Y, (too hard to try and write here, but it's just the pdf of x multiplied by the pdf of Y, since independent, where both are the pdfs of a standard normal). I don't know where to go from here... I know the joint mgf of two random variables A and B, M(s,t), is the integral over both a and b of the joint pdf of A and B multiplied by exp(as+bt). But I don’t know how to do that for U and V

I know that X=(Z1^2 + Z2^2 + ...Zn^2 ), and that Z~N(0,1), but I don't know where to go from there

Hi I'd be really grateful for any help on this.

I'm confused about power. I've read it should be 0.8 at least to ensure the avoidance of type 1 error (think it's type 1, I forget which). Anyway. If I do a chi squared, find a statistically significant difference, and then calculate power afterwards , if power is not >= 0.8 should I reject the null hypothesis or not?!

Them you so much

We're doing chi-square tests in high school statistics, and it is my understanding that the chi-square distribution is always skewed depending on the degrees of freedom. Why is this so? And how is the chi-square distribution found? And finally, why do we use a skewed distribution rather than a normal distribution that we use for other signfiicance tests?

Thanks in advance!

n is the number of categories

k is the number of estimations of population parameters that we make using our sample statistic (In this case, we estimate the population mean with the sample mean, so k = 1)

I understand that the degrees of freedom is the minimum number of independent variables required to specify the state of a system/ the number of independent variables which can vary freely.

What i don’t see is why estimating the population mean decreases the degrees of freedom by one.

Trying to get a better understanding of how chi-squared tests are reported (what information should be included). I understand how to compute a chi-squared test (a hypothesis should be written, then calculate an appropriate test statistic, find the p-value, then reject or fail to reject the null hypothesis, an state the results). But generally speaking what information should absolutely be included when reporting the results? Aside from addressing the hypothesis, what other information is expected to be included?

Hello, currently a undergrad stats major at my university, in a probability class. We just learned a whole host of probability distributions today, gamma, exponential, beta, chi square. And I came here to ask a question to clarify the relationship between them.

To my understanding and the way I was thinking about it was.

Gamma distribution, is skewed and takes up values which are in the set of Reals (0 to infinity). Only positive values. Has its respective variance and expected value

Exponential, chi square, and beta are all TYPES of a gamma distribution, and that a gamma distribution is like a overarching baseline for the three of them. But each had minor differences.

Exponential is parameterized strictly with a = 1 and B. So B will be most likely to change but alpha will always stay. Is also from 0 to infinity. Had a memory less property (don’t really understand why it’s for this and not others)

Beta, is strictly over the interval 0 to 1, parametized by alpha and beta

Chi-square, modeled by a gamma but has some constant v? Dont really get this either.

So could someone maybe fill in the gaps in my understanding here? Specifically

How do the 4 relate to each other?

What is the memory less properly in relation to the exponential distribution?

Why does chi square have this letter “v”?

Thanks

Currently taking a stats class at university, and although I know how to do all the calculations and utilize technology and the equations for each section, but I often struggle with the application side, or the explanation of purpose for such equations (Just trying to pass the class with an A, haha).

This question came, up and besides the obvious differences between the two, any idea what it's really asking me to explain? Because reading over the textbook doesn't give me anything really. (This is a 0 teach interaction class as well)

Question:

Give 2 characteristics of  the Chi-Square Distribution that are different than the normal distribution.