# Tag Info

### A and B are independent. Does P(A ∩ B|C) = P(A|C) · P(B|C) hold?

No this is not in general true, as you can see from a simple counter example: Toss two independent coins. Event $A$ is coin 1 head. $P(A)=0.5$ Event $B$ is coin 2 head. $P(B)=0.5$ Event $C$ is either ...
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### Probability that X > Y when X ~ N(0,2) and Y ~ N(0,1)

Your reasoning seems right. You can always do some simulation to check if your answer is correct. ...
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### Are SHAP values potentially misleading when predictors are highly correlated?

Summary Yes, SHAP values are potentially misleading when predictors are correlated -- they can be imprecise and even have the opposite sign. The correlation does not need to be incredibly high, around ...
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### Different amounts of observations for individuals - Which model to use?

As I mentioned in the comment, mixed/hierarchical/multilevel models are the general way to address this form of non-independence. The model is modified to deal with the correlations between points ...
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### Show that the two random variables with F-distribution are independent

Your answer to (a) is correct. For part (b), the two random variables are $F$-distributed by construction. We could prove that they're independent by establishing joint independence of $Y_1, Y_2, X_3$....
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### Prove that two random variables are independent

Conditioned on $X$ having value $x$, the distribution of $Y$ is $N(x,x^2)$. The conditional distribution of $Z = \dfrac YX$ given that $X=x$ is the same as the distribution of $\dfrac Yx$ which, as ...
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### Different amounts of observations for individuals - Which model to use?

If your sample data is representative of what your actual data will look like, then I don't think multilevel models are the way to go. First, most of your people have one observation and only one has ...
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### Statistical test for nominal data, multiple variables and within-subject design in R

You have a binary response (osteophytes), for which logistic regression is a good general approach. You have non-independence due to multiple measurements on the ...
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### What is the conditional $\operatorname{Var}(XY|Y)$ given that $X$ and $Y$ are independent?

Your reasoning flow shows that you had a good understanding on conditioning. Formally, you can derive it from the definition$^\dagger$ of conditional variance and basic properties of conditional ...
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### Representation of two Gaussian vectors as sums of independent Gaussian vectors

Let $X$ be $n$-dimensional, $Y$ be $m$-dimensional, and both with zero means. (It's simple to deal with nonzero means later because they just get added in.) We seek an $n$-dimensional zero-mean ...
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### Conditional distribution $f(x|y)$ if $X$ and $Y$ are independent

Yes, if the conditional density does not depend on the input $y$ then that is sufficient for independence. To see this, suppose that we can write $f(x|y) = h(x)$ for some function $h$ (i.e., as a ...
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### Probability that X > Y when X ~ N(0,2) and Y ~ N(0,1)

I'd like to add another simulation implemented in R. First, we can define PDFs for either random variable: ...
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### Does strict monotonicity imply image variable is dependent with domain variable?

No -- there are simple counterexamples. I take it that $X:(\Omega,\mathfrak F, \mathbb P)\to (E,\mathcal E)$ is a (generalized) random variable, that $E$ is endowed with a partial order $\le,$ and ...
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### Chi squared and Cramer's V as a measure of independence

I think your confusion comes from several things. In the first place, your calculations are theoretically correct, but apply to a two-way contingency table. The rub here seems to be that it looks like ...
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### Are SHAP values potentially misleading when predictors are highly correlated?

By default, TreeExplainer in the SHAP (SHapley Additive exPlanations) library uses feature_perturbation = "interventional". This choice is based on the ...

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### Case when random variable $X$ and its square $X^2$ are independent

I want to add that a sufficient and necessary condition for $X$ and $X^2$ are independent is that $X^2$ is degenerate. The sufficiency is trivial. Conversely, suppose $X$ and $X^2$ are independent. ...
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### Calculate joint distribution from marginal distributions

Yours is a particular case of the following theorem. Theorem. Let $X\,\sim\, \text{N}_p(\mu, \Sigma)$, $\underset{q\times p}{A}$ a and $\underset{q\times 1}{c}$ a fixed matrix and a fixed vector, ...
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### Can you combine effects from both the composite and domain levels of questionnaires in a meta-analysis?

I would say, yes, one can do this. However, I would not include subdomain level results and the composite result from the same study, since most 'composites' are just sum scores or means based on the ...
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### Independence across observations

Consider this counterexample showing that independence can hold without random sampling: Let $N_1, N_2 \sim \mathcal{N}(0, 1)$ independently, and let $X_1 := N_1$ and $X_2 := \beta X_2 + N_2$. Given ...
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### Independence across observations

Short answer: No. Longer answer: Random sampling is a different thing from independence. Take the archeypical sample of "sophomore college students in a survey course". This is not a random ...
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### Can we conclude from $\DeclareMathOperator{\E}{\mathbb{E}}\E g(X)h(Y)=\E g(X) \E h(Y)$ that $X,Y$ are independent?

This is almost Proposition 7.1.3(i) from Athreya & Lahiri 2006 sans some minor differences in formatting: Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\{X_1, \ldots, X_k \}$, \$2 ...
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### T-test with multiple measures per subject

No, probably not OK. How large an effect would it be reasonable to suppose that a green card would have on reported happiness? I would be surprised if it caused even a minor change and that means that ...
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