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).
2,525 questions
0
votes
0
answers
14
views
Conditioning on X to obtain distribution of parameter estimators in simple linear regression [duplicate]
Consider the simple linear regression model with the following assumptions:
I am trying to verify that $\dfrac{\hat{B}_1 - B_1}{\sigma / \sqrt{\sum_{i=1}^n (X_i - \bar{X})^2}}
\;\Big|\; X_1,\ldots,...
- 135
4
votes
2
answers
115
views
Show that $X$ and $f(Y)$ are conditionally independent given $Y$
Let $(\Omega, \mathcal{A}, P)$ be a probability space and let $X:(\Omega, \mathcal{A})\rightarrow (\Omega_1, \mathcal{A}_1)$, $Y:(\Omega, \mathcal{A})\rightarrow (\Omega_2, \mathcal{A}_2)$ and $f:(\...
- 975
5
votes
1
answer
142
views
Can conditioning eliminate VC dimension dependence in empirical process bounds?
I'm analyzing the function class:
$$
\mathcal{F} = \left\{ (x, z, y) \mapsto \mathbb{1}\{ y \leq z\alpha + x^\top\beta \} : \alpha \in \mathbb{R}, \beta \in \mathbb{R}^d \right\}.
$$
Let $\mathbb{G}_n(...
- 724
11
votes
2
answers
435
views
How to analytically sample from the conditional distribution of a t-statistic under normal data-generating process?
I'm working with a sequence of i.i.d. observations
$$
X_1, X_2, \dots, X_n \sim \mathcal{N}(\mu, \sigma^2),
$$
where both $\mu$ and $\sigma^2$ are unknown.
Define the studentized mean (i.e., the t-...
- 724
1
vote
0
answers
87
views
Definition of mean-preserving spread
I have the following definition for a mean preserving spread.
If $F,G$ are distributions on $\mathbb{R}$, then $G$ is a mean-preserving spread of $F$ if
There exists random variables $X,Y$ such that ...
- 391
4
votes
1
answer
115
views
conditional distribution and sufficiency
The question is related to
Puzzled by the definition of sufficient statistics in Mood, Graybill, and Boes
For a random sample $X_1, X_2, X_3, \dotsc, X_n$ from a distribution $f( ;\theta)$, a ...
- 73
2
votes
0
answers
47
views
Combining conditional probabilities without conditional independence
I am interested in finding P(A|B ∩ C) when I have P(A|B) and P(A|C). This question has been answered under the assumption that B and C are conditionally independent given A. My question is two fold:
...
- 21
1
vote
2
answers
207
views
Can conditional logistic regression be used when the number of events per stratum is not decided?
I'm trying to understand if I should use conditional logistic regression in non matched data. If so, why? What information is lost? Or what assumptions are we making?
When looking at the likelihood, I ...
- 3,884
4
votes
1
answer
91
views
Beta Binomial distribution conditional on realized gamma value
I am looking to derive the formula for the conditional probability distribution of a beta-binomial random variable $W$ conditional on value of a gamma random variable that drives the beta. The ...
- 41
2
votes
1
answer
210
views
Conditional Expectations and Expectations of Products of RVs
I have a pretty simple question. I just wanted to make sure I understand conditional expectations correctly. In particular if $X$ is a discrete random variable and $Y$ is continuous, we can write:
$$
...
- 335
3
votes
2
answers
179
views
How does one expand joint probability of 3 variables with one variable fixed?
I am brushing up on conditional probability and ran into these lecture notes. However I got stuck on the highlighted portion below under the section, Conditional independence.
Could you clarify why ...
- 153
0
votes
0
answers
48
views
Conditional distribution of sample mean when observations are removed at random
Suppose we have a sample of $n$ i.i.d. random variables $\{X_i\}$ with expectation $\mathbb{E}[X_i] = 0$ and variance $\mathbb{V}[X_i] = \sigma^2$. Assuming $n$ is ``large'' such that the CLT is ...
- 31
0
votes
0
answers
65
views
What does $\int_{0}^{t} [1 - G_2(c|x)-g_2(c|x)] \ h_1(x) \ dx$ mean in probabilistic terms?
Suppose $F(\cdot \ ,\cdot)$ is a distribution function and $f(\cdot \ , \cdot)$ is its density function. Let $g_i(x_i | x_j)$ and $h_i(v_i)$ denote the conditional and marginal densities derived from $...
- 1
2
votes
0
answers
112
views
Two Versions of the Hammersley Clifford Theorem
While studying Monte Carlo methods, I learned that the full conditionals $P(x_j \mid x_1, \ldots, x_{j-1}, x_{j+1}, \ldots, x_p)$ determine the joint distribution under some conditions. This result ...
- 21
3
votes
1
answer
119
views
Is $p(A|B) = p(A|f(B))$?
Let $A$ and $B$ be random variables and $f:\mathbb{R} \to \mathbb{R}$ one-to-one. Then, is $p(A|B=b) = p(A|f(B)=f(b))$? Intuitively, it seems that $f$ can neither destroy nor create our belief in the ...
0
votes
0
answers
50
views
Why is there a single conditional copula in a Vine copula model?
Say that I have 3 variables, X1 X2 and X3 and I want to fit a vine copula model to this dataset. In R, using the package VineCopula, I can do this using the function RVineStructureSelect. With 3 ...
10
votes
4
answers
1k
views
How to understand structure of sentences in probability
Here in my textbook, it says
(I) A randomly selected high school senior eats breakfast
(II) A randomly selected teenager is a high school senior who eats breakfast
(III) A randomly selected teenager ...
- 203
1
vote
0
answers
118
views
Multivariate conditional density when parametrized
In Schervish (1995), Definition 1.22 states
Let $(S, \mathcal{A}, \mu)$ be a probability space, and let
$(\mathcal{X}, \mathcal{B})$ and $(\Omega, \tau)$ be Borel spaces. Let
$X: S \rightarrow \...
- 81
7
votes
1
answer
333
views
Law of the unconscious statistician for conditional density
According to this answer, the law of the unconscious statistician implies
$$E[h(X) \mid Z=z]=\int h(x) g(x\mid z) ~\mathrm{d}x.$$
But what about
$$E[h(X \mid Y) \mid Z=z]?$$
Is it true that
$$E[h(X \...
- 81
3
votes
2
answers
233
views
If $X$ conditional on $f(Y)$ is Normal, then $X$ conditional on $Y$ is Normal?
Consider two real-valued random variables $X$ and $Y$ with full support. Let $f:\mathbb{R}\rightarrow \mathbb{R}$. Assume that
$$
X\mid f(Y) \sim N(\mu_{f(Y)}, \sigma_{f(Y)})
$$
That is: $X$ ...
- 1,016
1
vote
0
answers
60
views
Difference between conditional probability and naive bayes classifier
I have this dataset
a
b
c
K
1
0
1
1
1
1
1
1
0
1
1
0
1
1
0
0
1
0
1
0
0
0
0
1
0
0
0
1
0
0
1
0
I'm trying to calculate the probability of
P(K=1 ∣ a=1 ∧ b=1 ∧ c=0) using a Naive Bayes Classifier
If I ...
- 11
2
votes
2
answers
133
views
End-Tokens are Required to make Ngram Models Proper
The standard bigram model, (for example defined here) defines a probability distribution over a corpus $V$ based on the following principles:
The marginal probability of a word $w$ is defined as its ...
- 93
15
votes
4
answers
1k
views
In a Frequentist setting, how are we able to condition on the null hypothesis being True/False?
Paraphrasing Casella and Berger (2002): A hypothesis test is defined by a null hypothesis $H_0: \theta \in \Theta_0 $ and an alternative hypothesis $H_1: \theta \in \Theta_0^c = \bar{H_0}$, where $\...
- 804
2
votes
1
answer
130
views
Proving mutual independence of $(X, Y, Z)$ implies independence of $X$ and $Y$ given $Z$
I want to prove or disprove the following result:
let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $X, Y, Z : \Omega \to \mathbb R$ be mutually independent, $(\mathcal F, \mathcal B(\...
- 123
2
votes
2
answers
118
views
What is meant by "When we construct conditional probabilities, the relative proportions of probabilities remain the same."?
In the course "Introduction to probability" John Tsitsiklis states the following:
"When we construct conditional probabilities, ..., the relative proportions of probabilities remain ...
- 494