All Questions
Tagged with density-function conditional-probability
67 questions
0
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22
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Determining Distribution for Conditional Probability
I have that the conditional probability density of $Y|X$ is as such
$f_{Y | X} \propto x^{y - 1}(1-x)^{n-y-1}\alpha^{n-y}\beta^{y}$
where $\alpha, \beta$ are constants in $(0, 1)$, $x$ is a random ...
- 13
0
votes
1
answer
35
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Conditioning once or twice?
Let's say we have two random variables $Z \in \mathcal{Z}$ and $X \in \mathcal{X}$ with joint density $p_{Z,X}(z,x)$ with respect to a base measure. The density is assumed to factor as
$$ p_{Z,X}(z,x) ...
- 311
4
votes
1
answer
543
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An impossible distribution
Some days ago another user posted a question which was something like this:
$$ A \sim U(0,4)$$
$$B \sim U(0,6)$$
$$A - B \sim U(-4,4)$$
The question was originally to find the distribution of A ...
- 304
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0
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48
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Converting an integral into a probability of some event
Suppose that $X_1, X_2, .....X_n$ are iid random variables from some continuous distribution $F$. Show that $$\int_0^{\infty}(1-F(s+t))f(s)ds=\mathbb{P}(X_1>X_2+t, X_2>0)$$
$$$$Consider the ...
- 217
2
votes
1
answer
234
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Questions about the conditional Radon-Nikodym derivative
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})$ ...
- 863
3
votes
0
answers
118
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Conditional Distribution of Multivariate Gaussian given Linear Inequalities
Consider a multivariate Gaussian $Y\sim\mathcal{N}(\mu,\Sigma)$ of dimension $n$. For fixed $c\in\mathbb{R}^n, A\in\mathbb{R}^{m\times n}$ and $c\in\mathbb{R^m}$, what is the conditional distribution ...
- 31
1
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1
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159
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Calculating PDF conditioned on event
I'm confused about problems where we calculate a PDF conditioned on an event.
Consider this simple problem:
We have two random variables, X and Y, X is uniformly distributed on [a,b], and Y is uniform ...
0
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0
answers
51
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Derive E[Y|X] when the joint probability is given
Now, consider joint density of $X, Y$ :
$$
f_{X, Y}(x, y)=\left\{\begin{array}{l}
\frac{1}{\pi} ; X^2+Y^2<1 \\
0 ; \text { Otherwise }
\end{array}\right.
$$
Derive $E(Y \mid X)$.
I know how to ...
- 25
1
vote
1
answer
26
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Would this way of evaluating this probability be correct?
Suppose I have a discrete variable $S_t$ and a continuous variable $X_t$. Further, suppose I wish to evaluate $P(S_t=s_t)$. Would the below derivations be correct?
\begin{align}
P(S_t=s_t)&=\int P(...
- 1,226
0
votes
1
answer
560
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Finding the conditional distribution from given normal distributions using Bayes' theorem
Background
This question is related to my previous question: Describing the measurement of a random variable as another random variable, but I've narrowed and clarified my question. I think I've ...
- 101
3
votes
1
answer
425
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Conditional probability mass function of number of Poisson random variable given their sum values
We have a discrete random variable $N$, and $X_1, X_2, ... X_N$ are i.i.d Poisson random variables with parameter $\lambda$. Denote $Y = \sum_{i=1}^{N} X_i$. What I want to know is:
If finding the ...
- 33
4
votes
2
answers
292
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Difference between $\mathbb{E}[Y|X]$ and $\mathbb{E}[Y|X=x]$
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X: \Omega \rightarrow \mathcal{X}$ and $Y: \Omega \rightarrow \mathcal{Y}$ be random variables.
I have two questions comparing the ...
- 863
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54
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Different formulations of the conditional expectation [duplicate]
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X: \Omega \rightarrow \mathcal{X}$ and $Y: \Omega \rightarrow \mathcal{Y}$ be random variables.
I have a question comparing the ...
- 863
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0
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120
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Conditional Density Of Independent Bernoulli Random Variables Given Their Sum
Let Yi's be m independent Bernoulli random variables with corresponding success probabilities pi's, and let S = sum of Yi's.
I am trying to figure out a way to find the given conditional probability, ...
- 1
1
vote
1
answer
90
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How does one reduce the Maximum Likelihood rule to a pairwise comparison of the probability densities?
I am studying an article, where the authors have discussed something about "the ML decision rule reducing to a pairwise comparison of the conditional PDFs" because the means and variances ...
- 79
1
vote
1
answer
241
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Conditional distribution using Gamma and Weibull
I'm trying to compute the conditional distribution of $X|Y = y$.
$X\sim Gamma(3,2)$
$Y|X = x \sim Weibull(2,x)$
I was doing this:
$f_{X|Y = y}(x) \propto f_x(x)\cdot f_{Y|X = x}(y)$
$f_x(x) = \frac{1}{...
- 69
1
vote
0
answers
132
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Conditional Density Function on Underlying Exponential
Problem Statement: Suppose $Y_1, Y_2,\dots, Y_n$ are a random sample from an exponential distribution with mean $\theta.$ Let $\displaystyle U=\sum_{i=1}^n Y_i.$ Find the conditional density function ...
- 6,039
1
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0
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54
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joint and conditional density function
Let $f(x,y,z)$ be the joint density function. I found a reference that this joint density can be written as
$f(x,y,z) = f_1(x|y,z)f_2(y|z)f_3(z)$
I'm wondering if there are alternative forms to ...
- 387
0
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0
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198
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decomposition of joint density of multiple variables
I'm considering an EM algorithm of correlated random effects model
$y_{it} = \beta x_{it} + \mu_i + \varepsilon_{it},(i=1,...,n;t=1,...,T)$
where $y_{it}$ and $x_{it}$ are observed, but $\mu_i$ and $\...
- 387
1
vote
0
answers
80
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Conditional Probability involving condition on two RVs
Suppose X,Y~exp(2) and are independent. Let W=X+Y.
How do I set up integrals to calculate the following:
f(W|X>Y)
E(W|X>Y)
Thanks!
- 21
1
vote
0
answers
133
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Finding an expression for the CDF of sum of two random variables $X, Y$ conditioned on the value of one variable $Y$: find $P(X + Y < c | Y = b)$ [closed]
This is related to my other question on renewal processes https://math.stackexchange.com/questions/3947852/renewal-theory-probability-of-residual-lifetime-gamma-t-x-conditioned-on-c
$X, Y$ are ...
- 153
2
votes
1
answer
99
views
Conditional Gaussian
Suppose I have
$$
\begin{align}
p(x_1) &= N(x_1; 0, 1) \\
p(x_2 \mid x_1) &= N(x_2; x_1, 1)
\end{align}
$$
How do I compute $p(x_1 \mid x_2)$? I know how to compute their product, giving $N\...
- 2,286
1
vote
1
answer
49
views
Why is $f_{Y|X}(y|x) = f_\varepsilon(y - g(x))$ for the regression model $Y = g(X) + \varepsilon$?
Suppose we have the model
$$
Y = g(X) + \varepsilon,
$$
where the errors are zero-mean and independent of $X$.
I read that the conditional probability density function $f_{Y|X}(y|x)$ of $Y$ can be ...
- 153
14
votes
2
answers
1k
views
Why are density functions sometimes written with conditional notation?
I keep seeing density functions that don't explicitly arise from conditioning written with the conditional sign:
For example for the density of the Gaussian $N(\mu,\sigma)$ why write:
$$ f(x| \mu, \...
- 1,617
-1
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1
answer
25
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how to calculate the probabilty to draw a X form a gaussian pdf
Given a gaussian distribution, we draw a $X$ from it. What will the probability of drawing that $X$?
1
vote
1
answer
291
views
Conditional probability density from probabilities
I am trying to understand conditional probablility densities in relation to the conditional probablilities. From the Measure-theoretic definition on Wikipedia, if $X$ and $Y$ are non-degenerate and ...
- 13
4
votes
1
answer
84
views
How to get a PDF which converts an already drawn sample to uniform [closed]
Suppose i have a large data pool with a particular PDF, $F(x)$, interval $[x,y]$ estimated from KDE of the datapool. I drew $N$ samples at random from that data pool and saw that their distribution is ...
- 237
1
vote
0
answers
14
views
bivariate conditional joint sensor model
I am struggling to find $P\left( V_t | z \right)$ from $P\left( V_t | z , V_p \right)$. Here $z$ and $V_p$ are independent variables.
...
- 11
0
votes
0
answers
547
views
Probability density of conditional multivariate distribution [duplicate]
We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning ${\boldsymbol Y}$ into
$${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\
{...
- 6,187
0
votes
1
answer
272
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Sampling posterior distribution of a function
I have the following problem: let's say I have a function $y=f(x)$. Let $f$ be defined for all $x$ but it it might not be invertible. Further assume $x \sim p(x)$ with some probability density $p(x)$.
...
- 103
2
votes
2
answers
178
views
Given distribution of $X$ and $X|Y=y$, is it possible to find distribution of $Y$?
What the title says!
My intuition is NO since in Bayesian statistics we typically specify the prior and likelihood, and from those two we can compute the posterior and so on. We can interpret $Y$ = ...
- 61
1
vote
1
answer
296
views
Sum of two continuous random variables
Let R1 and R2 be two independent random variables, both with uniform density at the interval (0,2).
What is the probability of R1>1 given that R1 +R2<2?
--
What I've tried:
I know that
$$
P(R1&...
- 439
0
votes
0
answers
17
views
Gaussian Distribution [duplicate]
Assume we have two continuous Normal RV "X" and "Y".
how can I show the conditional PDF f(X|Y) and f(Y|X) is Normal?
0
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0
answers
56
views
Conditional density under conditional indepencence?
Let $X,Y,Z$ three random variables such that the joint density can be factorized as
$$f(x,y,z) = f(x \mid z) f(y\mid z) f(z).$$
This is, I am assuming conditional independence of $X$ and $Y$ given $Z$....
- 1
1
vote
1
answer
498
views
Computing a marginal distribution of a joint involving a delta function
Suppose that we have four continuous random variables $x,y,z,$ and $v$ and we want to compute the following integral:
$$\int f(x\mid y)f(z\mid x,y)f(v\mid z,x,y)\,dx$$
There are a few conditions:
$...
- 286
1
vote
0
answers
536
views
Changing a conditional probability to a deterministic function
Suppose that we have a conditional density function $p(y|x;\theta^*)$, where $\theta^*$ represents distribution parameters and are assumed to be deterministic. Is it possible that we write this ...
- 286
3
votes
1
answer
785
views
pdf from a set of conditional pdfs
I have an interesting problem, i have seen in many text books ways of calculating conditional pdfs but not many where given a set of conditional pdfs for a variable we wish to calculate it's pdf. In ...
- 477
0
votes
1
answer
179
views
Compute $E(X_1|X_1+X_2)$ $X_1, X_2$ both iid $Exponential(1)$
I recently stumbled across this question on CV:
Conditional expectation conditional on exponential random variable
And really liked the answer provided by @Rush, but I wanted to try to compute this ...
- 297
1
vote
2
answers
128
views
Condition Probability - What am I doing wrong?
I am reviewing some notes of mine and refreshing myself with some statistics and I came across a problem that asks me to calculate $P(X>1|Y>1)$ for the random variables $X$ and $Y$ whose joint ...
- 297
1
vote
1
answer
162
views
Credibility evaluation - how to model conditional continuous density from multiple variables of various types?
I recently got dataset for 37000 households with declared income and a few dozens of other variables of various types: continuous, discrete, binary.
The task is to automatically (unsupervised) ...
- 431
2
votes
3
answers
209
views
what does p( y | μ,σ²) really mean?
Just started to study Bayesian Statistics. I am very confused the concept of having a conditional probability on a distribution. Specifically:
I understand what p( A | B ) where A="I am sick" and ...
- 51
0
votes
1
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143
views
How do I estimate multiple probabilities of multiple values using a conditional multivariate norm in R? [closed]
I am using the condMVNorm package. I can get the density estimate for 1 value as follows.
...
- 1,400
1
vote
0
answers
392
views
How to integrate out a continuous r.v. from a mixed joint density
Say I have a continuous variable $C \sim N(0, 1)$, which was observed in a few points:
obsC <- seq(-1.5, 1.5, by = 1)
I also have a discrete variable $D \in \{...
- 2,617
1
vote
1
answer
317
views
Derivation of the conditional probability from an mixture of two normal distributions
I need to estimate the probability that an outcome R comes from either the sum (SUM) or from the difference (DIF) of two independent random variables. To do that, I need to compute the conditional ...
- 107
11
votes
5
answers
9k
views
Compound Poisson Distribution with sum of exponential random variables
I'm trying to find the distribution and parameters in a Compound Poisson
$S=\displaystyle\sum_{j=1}^{N}Y_{j},$ where $Y_{j}$ are exponential random variables independent and distributed identically ...
- 111
2
votes
0
answers
806
views
min/max and probability distributions
We have two identically distributed, independent, uniform variables on interval $[0,1]$ : $X_{1}$, $X_{2}$. And $Y_{1}=\max(X_{1},X_{2})$, $Y_{2}=\min(X_{1},X_{2})$. I want to find distribution $f(y_{...
- 273
1
vote
0
answers
17
views
conditional PDF bivariated Birnbaum Saunders
According to theorem 3.1 of this article here, If $(T_{1},T_{2})\sim BVBS(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2},\rho),$ then:
The conditional PDF of $T_{1},$ given $T_{2}=t_{2},$ is given by:
$$...
- 148
12
votes
3
answers
4k
views
How is $\Pr(X=x|Y=y)$ defined when $Y$ is continous and $X$ discrete?
Say that $Y$ is a continuous random variable, and $X$ is a discrete one.
$$
\Pr(X=x|Y=y) = \frac{\Pr(X=x)\Pr(Y=y|X=x)}{\Pr(Y=y)}
$$
As we know, $\Pr(Y=y) = 0$ because $Y$ is a continuous random ...
- 2,731
7
votes
0
answers
702
views
Spinograms vs. conditional densityplots
I have a binary response variable (hail) and multiple continuous predictor variables. My aim is to understand the linear/non-linear relationship of the predictors to the response to be able to justify ...
- 511
4
votes
1
answer
692
views
Gibbs sampling and conditional distribution
I need to simulate the posterior distribution of intraclass correlation coefficient $\pi(\rho|y)$ where $y$ is the data set and $\rho=\frac{\sigma_a^2}{\sigma_a^2+\sigma_e^2}$ with $\sigma^2_a\sim IG(\...
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