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4 votes
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Product of two independent Student distributions

When $X$ and $Y$ are independent random variables with densities $f_X$ and $f_Y,$ the density of their product can be found with a change of variables as $$f_{XY}(z) = \int_{\mathbb R} f_X(x) f_Y(z/x)\...
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3 votes

Why Can't We (Directly) Simulate Data From a Semi-Parametric Distribution?

The point is that the Cox "model" is written simply as: $$ \log \lambda(t|X=x) = \log \lambda_0(t) + x^T\beta $$ where $\lambda_0(t)$ is the baseline hazard function. It's like an intercept ...
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3 votes

Why Can't We (Directly) Simulate Data From a Semi-Parametric Distribution?

A stronger statement is true. It's not possible to simulate from any model! Simulations must draw from a distribution, and a model is a set of distributions. In order to simulate from a distribution ...
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  • 502
1 vote
Accepted

Can $ f(c | a, b) \propto f(a | b, c) $? When?

In general, if $a$ and $b$ are independent, you have: $$\begin{align} f(c|a,b) &= \frac{f(a,b,c)}{f(a,b)} \\[6pt] &= \frac{f(a,b,c)}{f(b,c)} \cdot \frac{f(b,c)}{f(a,b)} \\[6pt] &= f(a|b,c) ...
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1 vote

Can you have a "PMF-PDF" Together?

This is possible. It is standard measure theory and is done the same way as if all the random variables are continuous or all are discrete: You have a probability space $(\Omega, F, p)$ and your ...
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  • 3,138
1 vote

Hazard function and survival analysis

If your plot is intended to be the probability density function for an event over time, $f(t)$ in the terminology of the answer from @Martin Georg Haas, then you technically have an improper ...
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1 vote

How to interpret coefficients for a binary DV in an OLS model and constant value meaningfully?

There are a few different ways to model binary outcomes with an OLS approach, but the basic idea is that the coefficients represent the change in probability that the dependent variable is 1, holding ...
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3 votes

Hazard function and survival analysis

Answer Given that your $\lambda(t)$ actually represents the instantaneous probability to die at time $t$ (the hazard), your function $F(t)$ computes the probability to die before a certain time $t$. ...
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1 vote
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Joint distribution of longest run and number of runs in a sequence of $n$ coin tosses?

For the special case of $p=0.5$, the distribution is described by restricted integer compositions (see this post). $$ F(n,k,w)=\sum_{j=0}^k(-1)^j\binom{k}{j}\binom{n-jw-1}{k-1} $$ where $n$ is the ...
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  • 311
1 vote
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Betting on a sample from a known distribution

The median minimizes expected absolute distance between observations sampled from the distribution and a single value. That is: $$ \underset{m}{\text{min}}\text{ }\mathbb E\left[ \left\vert X - m\...
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0 votes

Estimate at which point a linear model hits a certain value (including probabilities)

The linear regression model in probabilistic terms is $$ \mu_t = \beta_0 + \beta_1 x_t \\ y_t \sim \mathcal{N}(\mu_t, \sigma^2) $$ so for any $x_t$, you can calculate the probability that $y_t$ is ...
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  • 113k
2 votes

Betting on a sample from a known distribution

The mean is affected by outliers in the distribution, so does not inform us about the mid-point of the distribution. The median is not affected by outliers. Consider three example distributions and ...
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  • 543
0 votes

Binary Classification Problem with Predicted Probabilities distribution skewed

It could be that you leaked data in some step of your process, but we hope to achieve such performance. It’s great to be able to look at a case (or have the model look at a case) and say, “Yep, that’s ...
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3 votes

MNIST with a TWIST, no labels given, only probabilities

By the way you’ve set up the problem, perfect accuracy is impossible, and we must accept that, even if we know a digit to be a $1$, there’s only a $90\%$ chance of being red, and that’s the best we ...
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5 votes

What is Galton's paradox?

The "paradox" here arises from sneaking in an implicit insinuation of independence that does not actually hold, which allows the argument to lead you to a wrong answer with a series of ...
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1 vote

What is meaning of such notations in general?

$$\mathbb{E}_{(\mathbf{u}, \mathbf{\sigma}, \mathbf{Y}_0)\sim N(0,1)\otimes \mu_D\otimes N(0,1)}\left[ e^{\mathbf{\sigma} t}\mathbf{Y}_0\mathbf{u}\right]=\int\int\int e^{\mathbf{\sigma} t}\mathbf{Y}_0\...
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  • 92k
3 votes

Probability of selecting 3 students being in different classes

This can be calculated more simply as: $$P(Friend1 \in any\ class) \times P(Friend2 \notin class_{friend1}) \times P(Friend3 \notin class_{friend1} \& Friend3 \notin class_{friend2})$$ $P(Friend1 \...
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  • 543
0 votes

Probability of selecting 3 students being in different classes

To do this in a straight forward way, friend1 can land in 1 of the 60 positions, 20 of which is in class1; then, friend2 can land in 1 of the remaining 59 positions, 20 of which is in class2; ...
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  • 101
6 votes
Accepted

In a game with 0.01 chance of survival, there are 100 participants, a specific player survives twice what are the odds?

The probability of a given player surviving twice (assuming survival is purely by chance and the two games are independent) is $.01^2 = .0001$, or 1:9999 odds. The probability of there being one ...
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  • 22k
1 vote
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Probability for a bin in a binned histogram

One style of histogram of a sample has a vertical axis called Density, scaled so that the total area of the histogram bars is unity $(1).$ Thus, suppose you have a large sample from a population with ...
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  • 50k
0 votes

How to calculate probability of winning from win rates?

I only see an easy solution if you assume that these winning probabilities are independent (which is only reasonable for a non-interactive game, where the two teams do not use their skills to ...
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  • 1
2 votes
Accepted

Fitting discrete data to continuous distributions

I recommend against using a continuous distribution to approximate a discrete distribution. Often it will work well enough, especially in data like yours where the counts are far from zero, but the ...
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  • 1,206
3 votes

Show that no two sets in the probability space with $\mathbb{P}(\{k\})=2^{-k!}$ are independent

The question outlines a rigorous proof -- but where does the idea come from? It all becomes clear when you write the probabilities in binary: from the binary representation of one of these ...
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  • 288k
3 votes
Accepted

Logistic regression simulation with respect to event occurrence (prevalence)

You have an array of explanatory variables $(x_1, x_2, \ldots, x_n)$ ($n=20000$) and a model that assigns a probability to each $x_i.$ You seek a subarray of these variables that has a mean ...
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  • 288k
0 votes

Logistic regression simulation with respect to event occurrence (prevalence)

Its going to be hard to simulate a with an exact proportion of 1s, but if you can get pretty close if you simulate a lot of data. Key thing to realize is that the intercept is the log odds of the ...
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1 vote

Consecutive coin flips, what is the appropriate statistical test for this word problem?

Heads to win, 1000 people flip coins, after 10 flips there is a winner every time No, this is not true. It is not every time. As you computed the probability for one or more winners is $100\% - 36.8\%...
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3 votes
Accepted

Prove $P(X_1+X_2> 2C) \leq P(X_1>C)$ if $X_1,X_2$ are identical, but dependent?

Counterexample $$P(X_1 = 3, X_2 = 7) = P(X_1 = 7, X_2 = 3) = 0.5$$ Then $$P(X_1 \geq 4) = 0.5$$ and $$P(X_1 + X_2 \geq 8) = 1$$ Thus in this case $P(X_1 \geq c) < P(X_1 + X_2\geq c)$ Below is a ...
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1 vote

$X_i, X_j$ independent when $i≠j$, but $X_1, X_2, X_3$ dependent?

One that's perhaps easier to think about comes from a chessboard. Pick a point uniformly on the chessboard and consider $X_1$: row number (1-8) modulo 2 $X_2$: column number (1-8) modulo 2 $X_3$: ...
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0 votes

Distribution function of the linear combination of standardized student-t quantiles

Answers are in McNeil, Frey and Embrechts, "Quantitative Risk Management", Princeton 2005. Seems that, assuming a student-t multivariate distribution, linear combination of marginals is also ...
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2 votes

Statistical Models that "Exploit" Distributional Knowledge of the Predictor Variables

In regression models, by definition, we are interested in the conditional distribution of the response variable $Y$ given the observed predictors (covariates) $X$. Namely, if the joint distribution ...
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0 votes

Probability of outcomes from overlapping samples of a random variable

I'm not completely satisfied with this answer, but I think it's better than no answer. I'll attempt to answer the question posed in the comments: in a 100-by-100 matrix of independent Bernoulli trials ...
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  • 311
3 votes
Accepted

What is the intuition behind the odds scale?

In the frequency interpretation, probability is the number of successful shots divided into the total number of shots (at each distance $x$). The odds is the number of successful shots per failure. ...
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2 votes
Accepted

probability that the players will exchange their initially drawn number

How about defining some cutoff value $q_1$ below which player 1 will decide to swap and a cut-off value $q_2$ below which player 2 will decide to swap. Then compute the win probability as a function ...
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0 votes

Notation of expectation with conditional in subscript

When you have a conditional expectation function, the input should be a function of the stipulated value of the conditioning variable. The conditional expectation is non-trivial only if the input ...
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  • 94.7k
0 votes
Accepted

Notation of expectation with conditional in subscript

The calligraphic "$\mathcal{X}$" describes the collection of all the $x_i$-values in your dataset, and the calligraphic "$\mathcal{Y}$" all the $y_i$-values. So, while both $\...
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2 votes

Conditioning of join gaussian over a line

Let $X$ and $Y$ be jointly normal random variables with means $\mu_X, \mu_Y$, and covariance matrix $\Sigma$. (We do not need that $X$ and $Y$ are independent, although it does simplify some ...
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2 votes

Conditioning of join gaussian over a line

A bivariate normal density can be likened to a piece of bologna (or did I mean to write baloney?) about which Americans often say "No matter how you slice it, it is still bologna". The ...
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0 votes

How to calculate real probability with multiple scenarios?

You need to calculate the payout ratio. From this case you 1 in 100 chance of drawing the \$100 bill so that average payout ratio is \$1. (win once, lose 99 times). So in this case with multiple ...
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  • 1,455
2 votes

MLE of the Uniform Distribution

In your example $n=3$, $\min x_i = 1$, & $\max x_i = 3$. When $\theta=1$, $$I(\max x_i \leq \theta, \min x_i \geq 0)=I(3\leq 1,1\geq 0)=I(\mathit{false})=0$$ This factor, & therefore the ...
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0 votes

What is the 'same distribution' mean?

Strictly speaking, it means that the CDF is the same. That is, the type of distribution, the mean, the variance, and all parameters are all the same, if they are well-defined. For most of the commonly ...
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  • 143
6 votes
Accepted

Is the probability of a continuous variable obtained via integrating over an interval of the probability density curve *cumulative* probability?

For easier reading, I have combined three extensive Comments (now deleted) into an Answer: You don't have the true PDF $f(x)$ from density in R. From the code, we ...
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  • 50k
0 votes

Looking for advice: Short-term forecasting using actual forecasts and real time data

The first step is to organise all of your historical data and clean it - you say you have inconsistent reporting due to network problems, this will lead to gaps in your historical dataset which will ...
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6 votes

Is the probability of a continuous variable obtained via integrating over an interval of the probability density curve *cumulative* probability?

The cumulative density function (cdf, $F(x)$) and the probability density function (pdf, $f(x)$) are related by the equation: $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(x)dx$ By the fundamental ...
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0 votes
Accepted

Finding probability of a team winning the title given probability of future matches

It's easy enough to enumerate all possible outcomes and sum up the probabilities of the outcomes where Man City wins the title. In R: ...
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  • 311
3 votes

Calculating the probability of guessing 3 card draws out of 6

In your example the first three guesses are right and the last three are wrong. But if the first three guesses were wrong and the last three were right, then you'd get a (slightly) different answer ...
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  • 2,726
3 votes

Calculating the probability of guessing 3 card draws out of 6

Hypergeometric distribution. The probability of drawing all three of three specified cards in six draws without replacement is a hypergeometric probability: $$\frac{{3\choose 3}{49\choose 3}}{{52\...
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  • 50k
1 vote

Estimate the value of a sigmoid function over expectation

No unbiased estimator exists, when $p(x)$ is Categorical distribution: Unbiased estimator of exponential of measure of a set? For the binomial distribution, why does no unbiased estimator exist for $1/...
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  • 117
0 votes

What does $\hat{\theta}_n = \theta + O_p(n^{-1/2})$ mean in terms of rates?

I like to view this intuitively in terms of quantiles. Then the notation for 'probability distributions' becomes exactly the same as 'regular' Landau-notation. The expression $P\left(|\sqrt{n}(\hat{\...
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What does $\hat{\theta}_n = \theta + O_p(n^{-1/2})$ mean in terms of rates?

It depends on what you are referring to with "the difference between $\hat \theta_n$ and $\theta_n$". Intuitively, it means that the larger $n$ becomes, the less likely it is that the ...
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  • 3,138
1 vote
Accepted

Fairness while counting chances

There is a reasonably well-developed physics/statistics literature on this topic There is actually a fairly well-developed literature on this matter in physics and statistics journals. Some good ...
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