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0 votes

Calibrating the probabilities of Ridge Classifier on imbalanced dataset

The problem is that the histogram of probabilities show that there is no separation between the 2 classes, with an almost normally distributed density With an AUROC of 0.76, I think this is expected. ...
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1 vote

Why use a copula to generate synthetic data?

I think the immediate goal of the problem is three-fold 1) to exhibit how a copula might arise when facing a data simulation task, 2) to get you using copula-based tools, and 3) to lead you to ask the ...
4 votes

Normalization of conditional probabilities

The answer by @CamilleGontier has pushed me in the right direction: what is implicit in the question (but was not explicitly stated in the OP) is that it should work for an arbitrary vector $p_i$ (as ...
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0 votes

What is the probability of multiple events occurring together?

Your problem seems to be ill-formulated. Your original formulation suggests that the employee left, and that you are computing the probability $p(X)$ that the event causing the employee to leave was $...
5 votes

Binary classification cross validation ROC score - only consider higher confidence class probabilities

When I do this, I lose about 85% of the test samples, but the resulting ROC score of the high confidence test set is boosted to 0.87, which makes it useful for downstream analysis. It sounds to me ...
5 votes

Binary classification cross validation ROC score - only consider higher confidence class probabilities

The ROC curve and corresponding AUC relate to assessing performance across a spectrum of thresholds, not performance at one particular threshold. Consequently, your plan seems to be equivalent to ...
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5 votes

Normalization of conditional probabilities

Not necessarily. Here is a counter-example: consider the vectors $$ p = \begin{bmatrix} 0.5 \\ 0.5 \end{bmatrix} $$ and $$ \pi = \begin{bmatrix} 0.5 \\ 0.5 \end{bmatrix} $$ ...
0 votes

What does conditional independence mean semantically?

It may be useful for you to initially think of 'dependence' and 'independence' here as statements about correlation or association, rather than as direct statements of causality. They are useful in ...
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1 vote

How to show for positive Borel functions $g, \int_{a}^{b} g(x)dF(x) = \int_{a}^{b} g(x)f(x)dx$

Albeit the concerns re the ambiguities as shown in the comments deserve clarification, these sort of problems are standard measure theoretic exercises. So, I am leaving below a general brief ...
1 vote

Is this a known measure of "effective degrees of freedom" in regression?

This quantity seems to reduce to "Welch-Satterthwaite degrees of freedom" is the limit of variance of $x,y$ going to 0
3 votes

Why is the probability of a set of outcomes possibly non-zero if each outcome's probability is zero in continuous probability?

What is this called and how is it resolved? It's called "lack of uncountable additivity". The property that for disjoint sets $A_i$ $$ P\left[\bigcup_{i\in S} A_i\right] = \sum_{i\in S} P[...
7 votes
Accepted

Why is the probability of a set of outcomes possibly non-zero if each outcome's probability is zero in continuous probability?

This is a consequence of the properties of integration, which is a large sum of infinitesimally small areas. When dealing with a continuous random variable $X$, the probability that it falls within ...
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1 vote

Why is the probability of a set of outcomes possibly non-zero if each outcome's probability is zero in continuous probability?

I can't really say much about the mathematics of infinitesimals and the probabilities of points in a continuous space, but a trivial answer is that even though the probability of a point in that ...
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1 vote

probability that the players will exchange their initially drawn number

I will solve with $U[0,1]$ as it is slightly nicer. Let $z$ be the cut-off for Player 1, under which he will want to swap, and similarly $w$ be the cut-off for Player 2. There are two scenarios: first ...
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1 vote
Accepted

Is the probability at least one out of four people flips heads on $\dfrac{3}{3}$ trials is $1-(\dfrac{7}{2^3})^4$ or $(1-\dfrac{7}{2^3})^4$?

The probability that one person do not find three heads is $7/8$. So the probability that nobody of the four find three heads is $(7/8)^4$. Therefore $P(A)=1-(7/8)^4$, then a bit more than $41 \%$. So ...
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1 vote
Accepted

How to show that $X_n + Y_n \to X + Y$ holds in the $L^1$ norm?

These are standard exercises. So, let me leave behind the ingredients that OP can utilise to formally construct the proofs. Let $(\Omega, \mathfrak A, \mathbf P) $ be the probability measure space. ...
2 votes

Strategy for game where larger number wins. Drawn from standard uniform distribution with one redraw allowed

Firstly, I will explain why maximizing the expected value does not work (BruceET's answer is wrong). Suppose the players play the following game with dice, whoever gets the larger number wins. Player ...
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1 vote

Is $Y=Y(\omega) = \inf_{0 \leq t \leq 1}X_t(\omega) = 1_A(\omega)$ not measurable if $A \notin \mathcal{B}[0,1]?$

Observation $1$. Let $(X, \mathfrak A) $ be a measurable space and let $A\subset X. $ Then $\mathbf 1_A$ is $\mathfrak A$-measurable if and only if $A\in \mathfrak A. $ The proof is straightforward ...
0 votes

Is it possible to use a scaled version of a beta distribution to represent lifetime?

To see why it’s not recommended, think about a Beta scaled to (0, 100). This would make everything below 100 (but positive) a live possibility, but everything above 100 completely impossible (along ...
1 vote

How do Measure "Robustness" in Statistics?

One method to examine "robustness" is to do a simulation analysis Sometimes we have a statistical model and its estimators for some type of data, and we want to see if the estimators are &...
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4 votes
Accepted

Gaussian fourth-moment formulas?

I have never seen an closed form expression for this. Probably because it is quite ugly. I have worked with a similar expression before, and I'd be happy to see if my expression is stands up to yours. ...
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0 votes

Is it possible to use a scaled version of a beta distribution to represent lifetime?

No, it is not. There are other distributions that "tell a story" that fits better with lifetime data. The simplest example would be the exponential that assumes a constant hazard. Other ...
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1 vote
Accepted

Frequentist method for random samples from unknown urn

Bayes' Theorem is just a simple identity in probability theory for two random variables that is independent of what your fundamental interpretation of statistics would be. So both a frequentist and a ...
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0 votes

Mathematical knowledge needed for learning upper level statistics

As a start to study Statistics mathematically and rigorously, I would recommend you to read: Shao J - Mathematical Statistics It is a very good book to begin your journey.
1 vote

Would there be any alternatives to a Logistic Regression or way to modify the Regression for what I'm looking for?

My data is widely dispersed across the X axis for both my 1s and 0s however there is slightly more 1s the higher the X and slightly more 0s the lower the X. So the probabilities listed in a standard ...
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4 votes

Would there be any alternatives to a Logistic Regression or way to modify the Regression for what I'm looking for?

The probability will not hit 0% or 100% exactly, it approaches them asymptotically. The uncertainty increases towards the extremes, which you will see if you include confidence intervals in your plot. ...
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4 votes

If $F_X, F_Y$ agree for all $x \in \mathbb{R}$, Do their distributions $\mu_X, \mu_Y$ agree on $\mathcal{B}$?

Observation $1.$ Let $\mathbf P_1, ~\mathbf P_2$ be two probability measures on $(\Omega, \mathcal F). $ Let $\mathcal P$ be a $\pi$-system such that $$\mathbf P_1(A) =\mathbf P_2(A), ~~~\forall A\in \...
0 votes

Modelling probabilities of a sum of binomials with different probabilities and trials

Let $A_i,B_i,C_i,Y_i$ denote the values in the $i$th row. If $p_A,p_B,p_C$ are the same in all rows, then the appropriate model is $$Y_i \sim \text{Bernoulli}(A_i,p_A) + \text{Bernoulli}(B_i,p_B) + \...
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1 vote

Is there a test that can tell me, even for low counts, the probability that the obtained distribution of values eg from a die is truly random?

Test for goodness of fit with categorical data, whether a particular distribution is a good fit for a particular sample, are the Pearson's chi-squared test and the G-test. Note that those tests are ...
1 vote

How to estimate the probability mass function of a discrete variable from moments

I have worked with this problem a lot, and I cannot find a satisfactory answer, but here are a few ways that I've attacked it. (1) If $M=k$, and if you know that a valid probability distribution can ...
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5 votes
Accepted

Proof of corollary of Hoeffding's inequality

With $$ \begin{align} & \mathbb{E}\left(\sum_{i=1}^nX_i\right) = n \cdot \mu, \\ & a_i=0, \\ & b_i=1, \\ & \tilde{\varepsilon} \mathrel{:=} n\cdot\varepsilon, \end{align} $$ we have $$ ...
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2 votes

Can we say the frequentist interpretation of probability is more appropriate in the dice rolling problem?

Bayesians and modern "frequentists" both use the same underlying laws of probability theory, and in my view, both can reconcile their views on the "frequentist definition of probability&...
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1 vote

KL divergence between gaussian and uniform distribution

How else can I calculate the distance of my gaussian to a 'maximum entropy' distribution if I can't use the uniform distribution? You basically answered your question, you can use the entropy of the ...
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1 vote

How to interpret height in probability density function?

You should start with the Can a probability distribution value exceeding 1 be OK? thread that explains the concept of probability density in detail. If we have f(a) = 2 * f(b), can we say that the ...
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2 votes
Accepted

Kolmogorov axioms consequences

the prof mentioned that there does not exist a probability measure P(A) defined for all subsets of [0,1] which would satisfy all the 3 Kolmogorov axioms At the very outset, please note whuber's ...
2 votes
Accepted

Can we say the frequentist interpretation of probability is more appropriate in the dice rolling problem?

If you know that the die is fair, then this can equally be viewed as a Bayesian interpretation with a dogmatic prior that puts all prior probability on the die being fair, so one could argue that the ...
1 vote
Accepted

Binomial distribution and common sense aren't adding up

The problem is that what you are referring to is not binomial distribution, but normal approximation of it. As you can learn from the Wikipedia article on binomial, the normal approximation does not ...
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1 vote

Why does central limit theorem give such big x in $\phi(x)$

The set up of the question is absurd, as originally pointed out by Michael M, in that the expected number with $200$ days is $10000$ with a standard deviation of $100$, so you are extremely unlikely ...
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1 vote
Accepted

Why does central limit theorem give such big x in $\phi(x)$

You're looking at a case where (assuming independence, though it would seem to be a somewhat questionable assumption), the distribution of the number of visitors in $200$ days has $\mu=10000$ and $\...
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2 votes
Accepted

Estimating the average building size

Suppose, the number of people living in a random building of the city (chosen uniformly) is $X$ and has probability mass function $P(n)$. Now the number of people living in the building where a random ...
0 votes

Determine if a categorical variable occurs at a frequency greater than chance with a boolean outcome

Let's start with the following (sloppy) hypothesis: $H_0$: Drug $A$ has the same probability of being prescribed when a patient has disease $C$ as for any other disease. Let's assume that sufficient ...
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1 vote
Accepted

What information can I extract from an overlap of two personal probability distributions?

The ratio of two gamma-distributed RVs is the generalized beta prime distribution. The CDF can be defined in terms of the ordinary hypergeometric function, ${}_2F_1$: $F(x;\alpha,\beta,q)=\frac{(xq)^{\...
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1 vote

Probability that next flip is tails, given $P(p ≤ z) = z^4$ + last five flips were tails

You have found the expected value of the prior probability of tails before any data were observed. $f_Z(z)=4z^3$ But after $X = 5$ tails after $n = 5$ flips, Alice has more information about $z$. From ...
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3 votes
Accepted

Confused with independence and Bayes theorem

Your second way is incorrect because $\prod_{i=1}^n p(x_i,\theta|\overline{y})\neq\prod_{i=1}^n p(x_i,\theta|y_i)$. Perhaps this becomes more clear if you try it for two variables: $$p(x_1,x_2,\theta|...
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-1 votes

Expected number after n rounds of uniform~[0,1] draws

I think you are overcomplicating this. With $X_1, X_2, \dotsc, X_n$ iid uniform on the standard interval $(0,1)$, your construction is simply computing $\max_{i=1}^n X_i$, and to find that ...
4 votes

Consistency of values generated with a Poisson inverse function and random numbers in Python

If you are surprised that a large proportion of your simulation runs have an absolute difference between wins and losses of more than $50$ games i.e. more than $\pm 0.5\%$ of the $10000$ games ...
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1 vote

Calculate the likely number of distinct categorical values (diagnoses) for a patient population

This is related to the Coupon collector's problem as noted in the comments. Building off of this post, the probability of observing $k$ unique letters in $m$ random samples from an alphabet of size $n$...
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0 votes

How to guess the size of a set?

First some notation: Assume $N$ unique words, sampled with the same probability. We are doing $R$ (independent) rounds of simple random sampling without replacement, each round the sample size is $n \...
0 votes

How to guess the size of a set?

Using the following notation: $s_i$: the number of samples for the $i^{th}$ round $k_i$: the number of words sampled in the $i^{th}$ round that had not been previously sampled $m_i=\sum_{j=1}^ik_j$ $\...
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3 votes
Accepted

Continuous and differentiable bell-shaped distribution on $[a, b]$

Let's construct all possible solutions. By "distribution" you appear to refer to a density function (PDF) $f.$ The properties you require are Supported on $[a,b].$ That is, $f(x)=0$ for ...
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