# Questions tagged [conditional-probability]

The probability that an event A will occur, when another event B is known to occur or to have occurred. It is commonly denoted by P(A|B).

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### KDE-like technique to learn a continuous distribution from samples subject to specific noise

There's a continuous-valued random variable $X$ with distribution $f_X$. Normally, we're given a bunch of i.i.d. samples $X_1, \ldots, X_n$, and we try to give an estimate $\hat{f}_X$ of the ...
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### What is the expected number of children until having the same number of girls and boys?

A couple decides to keep having children until they have the same number of boys and girls, and then stop. Assume they never have twins, that the "trials" are independent with probability 1/...
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### Two notions of independence?

I would like some clarification as to if independence as referred to in both of the following cases, is the same or different. I would assume they are different? Scenario 1: For any SINGLE trial of a ...
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### Interview question - Over the long term what fraction do I expect to spend in the left lane?

The following is an interview question: A certain highway has two lanes in my direction. Whenever traffic moves, if I'm in the left lane I switch to the right with probability 60%. If I'm in the ...
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### Cumulative distribution of Gaussian conditional independent random variables

Suppose X, Y, Z are three jointly Gaussian random variables and X and Z are independent given Y. For example, take three r.v. from a OU process. Here is some R code:...
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Most problems about martingales start with an assertion similar to the following one: Suppose that $X_n$ are iid such that $\mathbb{E}[|X_n|]<\infty$ and $\mathbb{E}[X_n]=0$. Then $S_n = \sum_i^n ... 1 vote 0 answers 35 views ### Semantics: Conditional Distribution vs Marginal Distribution This is something which has always confused me. Suppose we take a standard statistical regression model: \begin{equation} Y_i = \beta_0 + \beta_1 X_i + \epsilon_i \end{equation} where:$Y_i$is the ... 0 votes 0 answers 24 views ### Probability of next flip being heads given I have seen h heads and t tails I am currently attempting to understand "Question 2" at this link but having many difficulties. The problem is as follows: A coin has a chance of landing heads with an unknown probability ... 0 votes 1 answer 35 views ### Why is this integral not Lebesgue measurable? (and how to rectify) Suppose I have the Bernoulli random variables$S_t$for$t=1,\cdots, T$and for an issue related to my problem, I want to express$P[S_t=s_t\mid S_{t-1} = s_{t-1}]$, by using the law of total ... 1 vote 1 answer 29 views ### Probability of heads given p is less than some value Say I have a coin with a chance$p$of landing heads where$p$is drawn from the uniform distribution from 0 to 1. Now let's say I have some fixed number$x \in (0,1)$. My question is what is the ... 0 votes 1 answer 18 views ### distributions in "A Note on the Inception Score" [closed] I'm reading the following paper A Note on the Inception Score. IS is a evaluation metric for GAN. While reading the paper I got very confused about the notation of so called distributions. I will ... 1 vote 0 answers 38 views ### Why:$y | x , \theta \sim N(\theta^T x , \sigma^2)[closed] Given: \begin{align*} &\epsilon \sim N(0, \sigma^2),\\\\ &\theta \sim N(0, \eta^2I), \\\\ &\text{And } y^{(i)} = \theta^T x^{(i)} + \epsilon^{(i)} \\\\ \end{align*} Why: y^{(i)} | x^{... 0 votes 1 answer 53 views ### What is going on? Contradictory results on the variance of random vector with random mean and covariance Suppose f\mid\mu, F\sim N(\mu, F) \in \mathbb{R}^n, where \mu , F are both random (random vectors and random matrix respectively). What is the correct way to derive Var(f)? First, let \tilde{f} ... 0 votes 1 answer 61 views ### Converting a Conditional Expectation into an Unconditional one Consider two random variables V_1 and V_2, where V_1 is continuous and V_2 is binary. In a paper, I found that the following equality holds: \begin{align*} \mathbb{E}\left[V_1|V_2=1\right]=\... 2 votes 1 answer 33 views ### How to correctly compute conditional probabilities by considering the probability of each possible outcome? So I've been trying to understand the boy-girl paradox, and I reframed it in terms of coin flips to help it make more sense to me and I get now why the probability is different depending on whether we'... 0 votes 0 answers 57 views ### Is a nested expectation equivalent to expectation on the joint? \mathbb{E}_{p(x)}[\mathbb{E}_{p(y \mid x)}[f(x, y)]] = \mathbb{E}_{p(x, y)}[f(x, y)] Is it always true that nested expectations equal expectations on the joint? \mathbb{E}_{p(x)}[\mathbb{E}_{p(y \mid x)}[f(x, y)]] = \mathbb{E}_{p(x, y)}[f(x, y)] $$Something along these lines ... 1 vote 1 answer 44 views ### What is the correct way of expressing this probability mass function? Suppose I have a discrete random variable S. Now further suppose that I have a random variable X which is a continuous random variable on the real line. I wanted to express the p.m.f. P[S=s] for ... 0 votes 0 answers 7 views ### Maxdiff Approach - Claims comparison , how to compare product claims of two different surveys Maxdiff gives Preference shares or count based analysis My business objective: I want to create a MAX diff approach where I will have multiple surveys my output will be Claims and its Posterior Probability , count of best and worst selection. For more ... 1 vote 1 answer 36 views ### Conditional CDF given one dimension equals derivative of joint CDF towards that dimension divided by the density at that dimension? So I am familiar with the following:$$P\left(X<x|Y=y\right) =\int_{-\infty}^{x}f\left(X=u|Y=y\right)du=\frac{1}{f\left(Y=y\right)}\cdot\int_{-\infty}^{x}f\left(X=u,Y=y\right)du$But during a ... 1 vote 1 answer 63 views ### Conditional distribution of product of two random variables given one of the random variables Let's say I have two independent random variables$X$,$Y$. Let's define a new random variable$Z = XY$. I know that the distribution of$Z$follows:$f_{Z}(z) = \int_{- \infty}^{\infty}f_{X}(x)f_{Y}(...
Given a random variable $Y \sim D$ and a linear transform $g(Y) = a + bY$, we define a new random variable $W = g(Y)$. The probability density of $W$ is $f_{W}(w) = \cfrac{1}{b}f_{Y}(\cfrac{w-a}{b})$. ...
Let $P_{XYZ}$ be the joint distribution of discrete RVs $X,Y,Z$ where $Z$ is binary-valued. Let $Q_{XY}=P_{XY|Z=0}$, i.e. the distribution of $XY$ conditioned on $Z=0$. Are there lower/upper bounds on ...