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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What's the maximum expectation of a conditional variance, $E[Var(X+Z_1 \mid X+Z_2)]$?

Let $X,Z_1,Z_2$ be 3 mutually independent RV's, with $Z_1, Z_2$ assuming $N(0,1)$ distribution. $X$ is constrained to have unit 2nd moment, i.e. $E[X^2] =1$, but may take arbitrary distribution. The ...
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1answer
13 views

Conditional probability in random variables [on hold]

Let the time (in hours) needed to repair a broken car be a continuous random variable $X$. Then, in the case of $a$ and $b$ being two positive numbers and $b$ > $a$, would the following two problems ...
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1answer
36 views

Closed form of conditional probability for a specific joint

I have a joint probability of a very specific form: $P(x_1,\cdots,x_n)=\phi(x_1)\psi(x_1,x_2)\phi(x_2)\cdots\psi(x_{n-1},x_n)\phi(x_n)=\prod_{i=1}^n \phi(x_i) \prod_{i=1}^{n-1} \psi(x_i,x_{i+1})$ I ...
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11 views

Relationship of instantaneous variance and VIX with conditional density functions

I read a paper about the VIX which contains a stochastic volatility model. To estimate the parameters of SV (to be later able to price VIX futures), they derived a conditional probability density ...
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21 views

How to combine posterior probabilities from different classifiers?

I have four different images $(X_1, X_2, X_3, X_4)$ which I classify with four different discriminate probabilistic models (discriminative classifiers) to obtain posterior probabilities of a pixel ...
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41 views

Conditional Probability question about gym members

at a particular gym, 70% of the members take at least one fitness class and work with a personal trainer. if 18% of the gym members work with a personal trainer, find the probability that a randomly ...
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29 views

Variation of the Coupon Collecting Problem

Suppose you are collecting coupons from a pool of 50 distinct types. These types can be divided into 10 sets of 5 types each. What is the expected number of coupons you need to collect in order to ...
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29 views

probability distribution of complex Gaussian column vector and conditional probability of complex Gaussian column vector

I have column vector $\vec r=[r_1\ r_2]^T$. $$\vec r =hA\vec s +\vec n$$ where $h$ is a complex number , $\vec n \sim \mathcal{C} \mathcal{N}(\begin{bmatrix} 0 \\ 0 ...
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1answer
116 views

Proof of Simplification of Conditional Expectation of Product of Random Variables

Could someone please provide detailed steps to prove or disprove the following? $E[XY\mid XY>k] = E[XE[Y\mid XY>k]]$ Here, $X,Y$ are independent random variables that could be discrete or ...
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1answer
22 views

Proof that Conditional Expectation of Sum is Sum of Conditional Expectations

\begin{eqnarray*} E\left[\left.\left(X+k\right)\right|\left(X+k\right)>0\right] & = & E\left[k\left|\left(X+k\right)>0\right.\right]+E\left[X\left|\left(X+k\right)>0\right.\right] ...
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1answer
31 views

Independence of random variables after a transformation

We are given four random variables $S:=(S_1,S_2,S_3,S_4)$ defined on $(\Omega, \mathcal A,P)$. The random variables can be viewed as being extracted from a stochastic process. I assume that all ...
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27 views

Conditional versus Unconditional MLE parameters estimation?

I have read about both conditional and unconditional MLE parameters estimation for ARIMA models but I have problem understanding the concept from the statistical books . Does conditional MLE mean that ...
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1answer
11 views

Hierarchical process of exponentials

I'd like to work with a what I believe is a called a "hierarchical process" -- given by the multiplication of a pair of exponential distributions such that the random variable from one process is the ...
1
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2answers
64 views

Process with parameters that are themselves statistical

I'd like to work with a pair of statistical processes such that the random variable from one process is the parameter of the second process. The simplest case I can imagine (and which is still ...
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8 views

Check which correlation is better when numerical values are same

I have a data-set words = c((a,b,d),(a,b,d),(f,b,d),(m,n,d),(k,l,d)) Now, I want to find out what is the probability of occurrence of b given a occurred i.e P(b|a) which comes out to be 1. ...
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13 views

Kullback-Liebler's divergence on a conditioned function

Let $q$ be a conditioned pdf over $\mathbf{X}=X_1,\dots,X_n$ binary r.v.s in the form $$q(\mathbf{X})=\begin{cases}q_{0}(\mathbf{X}_{\setminus i}) \text{ if } X_{i}=0\\q_{1}(\mathbf{X}_{\setminus i}) ...
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1answer
30 views

Conceptual proof that conditional of a multivariate Gaussian is multivariate Gaussian

I understand the arithmetic derivation of the PDF of a conditional distribution of a multivariate Gaussian, as explained here, for example. Does anyone know of a more conceptual (perhaps, co-ordinate ...
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25 views

Joint probability for non-mutually exclusive events

90% of people who like bananas eat a lot. 60% of those who like apples eat a lot. If person "A" likes banana and likes apples, what’s the probability that they eat a lot. Is this possible to solve?
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27 views

Bayesian decision theory - loss function

i am trying to learn bayesian decision theory. the book that i am using (Kevin Murphy: Machine Learning: A Probabilistic Perspective) - has the following introduction on the action a: however - ...
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2answers
30 views

Bayesian Inference and Conditional Probabilities

From Wikipedia, under "Formal description of Bayesian inference:" $\theta$, the parameter of the data point's distribution, i.e., $x \sim p(x|\theta)$ $\alpha$, the hyperparameter of the ...
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3answers
82 views

Proving Berkson's Fallacy

This is a question based on Berkson's Fallacy. In particular, does the following inequality hold true: $ P(A | A \cup B ) \geq P(A) $ If so, how does one show it?
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1answer
9 views

Conditional probability of % change in stock A given % change in stock B

To start off, I've been reading "Statistics: Principles and Methods" 7th edition by Johnson and Bhattacharyya. I understand the conditional probability formulas and have practiced the examples in the ...
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12 views

Maximal entropy distribution with given conditionals

It is well known that of all the joint distributions $p(x,y)$ with fixed marginals $p(x),p(y)$, the one with the highest entropy is: $$ p(x,y)=p(x)p(y). $$ Suppose instead that we have conditionals. ...
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182 views

Bayesian regression full conditional distribution

I have a problem with the derivation of the full conditional distribution of the regression coefficients in a simple Bayesian regression. The source of the following equations is: Lynch (2007). ...
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1answer
95 views

Why does drawing one card at a time increase the probability of choosing the Ace of Spades?

If you draw 5 cards from a standard deck of 52 cards, then the probability of your hand having the Ace of Spades is: $$\frac{51\choose 4}{52\choose 5} = \frac{51!5!47!}{4!47!52!} = \frac{5}{52}$$ If, ...
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9 views

Connecting vertices in a graph stochastically. How to calculate a joint condition?

Full disclosure: I've posted this on reddit at /r/askstatistics, but haven't gotten any feedback yet, so I'm re-posting here in the hopes of getting some more exposure. Sorry for the long question. I ...
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32 views

Normalize estimated probabilities from two logistic regression models

I have built two logistic regression models predicting the probability of purchase of two products. (Product A and Product B) For every customers, I want to choose the product that has the higher ...
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22 views

Help Simple Conditional Counts Example

Let's say we have a sequence $S$ \begin{align} t \quad 0 \quad 1 \quad 2 \quad 3 \quad 4 \\ S_t \quad 1 \quad 1 \quad 0 \quad 0 \quad 1 \end{align} And we want to predict $S_{t+1}$ by selecting ...
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30 views

Confusion about joint and conditional probability

A person has 16 headaches (H) in 30 days, means P(H) = 16/30 = 0.53 The person tracked his Stress Level and Lack of Sleep for those 30 days. It is assumed that High Stress (HS) and Lack of Sleep ...
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1answer
23 views

Order Statistics Conditional Distribution of Affiliated System

We have a system with $M (M\ge 2)$ random variables. The M variables are related as follows. For each i, 1 to M, $X_i = I_i+Z$, where $I_i$, Z are independent uniform random variables. What is the ...
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1answer
78 views

How to calculate joint probability based on result of two conditional probabilities?

I have P(Headache|Stress) = 0.88 where Stress is a "necessary" cause. Again, I have P(Headache|Coffee) = 0.35 where Coffee is also a "necessary" cause. Now I want to calculate joint effect of Stress ...
2
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0answers
30 views

Conditional probability with multiple dependent conditions

A set of items is said to be incorrect if at least one item in the set is incorrect. Initially, all items have equal probability of being correct/incorrect (P = 0.5). A series of infinite events ...
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66 views

Conditional PMF for $P_X(x) = \frac{2(x - a + 1) }{(b-a+1)(b-a+2)}$

The PMF of random variable $X$, is given by: $$P_X(x) = \begin{cases} \frac{2(x - a + 1) }{(b-a+1)(b-a+2)} & a \leq x \leq b \\ 0 &\mathrm{otherwise}. \end{cases}$$ Assume $a < 0$ and ...
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3answers
466 views

Conditional Expected Value of Product of Normal and Log-Normal Distribution

Could someone please provide the answer and steps to solve this expression? \begin{eqnarray*} E\left[\left.\left(e^{X}Y+k\right)\right|\left.\left(e^{X}Y+k\right)>0\right]\right. \end{eqnarray*} ...
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43 views

How is free energy an unnormalized conditional log-probability?

I am following Bengio's Learning Deep Architectures for AI and at page 28 there is a phrase that confuses me: $a(x)$ is the discriminant function or an unnormalized conditional log-probability, ...
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0answers
18 views

Computing covariance between normal and uniform distributions

I think that this question is related: Covariance of a compound distribution, but it's too abstract for me (I believe that what I have to do must be simpler). Here's the given: machines $A$ and $B$ ...
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2answers
60 views

Bayesian Network Problem: What is P(A=T, B=T, C=F, D=F)?

Suppose you have these probabilities in a bayesian network. What is P(A=T, B=T, C=F, D=F)? I attempted to answer this by saying that P(A,B,C,D) = P(A) P(B|A) P(C|A,B) P(D|A,B,C), according to the ...
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1answer
32 views

how to calculate the following conditional probability

There are two events involved, say event A and event B. I want to know the probability of event B conditioned on the event A. The relation between the two events are as follows. We can not talk about ...
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3answers
73 views

Conditioning on vs. fixing a random variable

I am confused by the following notation, seen used by a professor in a course I'm taking. $p(X|Y)$ denotes the conditioning of a distribution over a random variable $X$ by $Y$ and $p(X;y)$ denotes ...
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176 views

Multiple conditional probabilities

Similar questions have been asked in this context but this one seems a bit different. [Aim] We would like to find out what the probability is of a person purchasing a 4USD column D (see below) price ...
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34 views

Example for conditional Poisson regression

I am working with the dataset below, I am trying to predict the chances of ...
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40 views

P(U | C) = ? where U = (A, B) where A and B are independent events

I want to calculate $P(C | A, B)$ or $P(C | U)$. I know $P(C | U) = \frac{P(U | C) \times P(C)}{P(U)}$. Since $A$ and $B$ are independent, $P(U) = P(A)\times P(B)$ But how to calculate $P(U | C)$? ...
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42 views
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40 views

Evaluating the conditional distribution $f_{Y|X}(y|x)$ without the joint distribution $f_{XY}(x,y)$

Let the distribution $Y = X + N$. Where $X$ and $N$ are independent and they have distinct distributions. I have $f_X(x)$ but I don't have the $f_{XY}(x,y)$ to use, for example, the following ...
3
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23 views

Calculating empirical distribution function from data

Suppose that we have a matrix (n lines and 3 column) which represents the value of three variable X, Y, Z at some instants : X Y Z x1 y1 z1 x2 y2 z2 ... xn yn zn Given that X ...
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1answer
54 views

Conditional density of bivariate normal and normal

Let $Z=X+Y$ where $X \sim N(\mu,\sigma^2)$ and $Y \sim N(0,1)$ are independents. What is the conditional density of X given Z, $f_{X|Z}(x|z)$? I already found that ...
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9 views

Conditional probablity of k-th order statistic of a column given the k-th order statistic of the sum of the columns?

Suppose A is a random matrix. Each row is a series of i.i.d random variables. I like to know if we can calculate the conditional probability (for a given $k$) $$P\big(A^i_{(k)} \mid (\sum ...
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1answer
85 views

Conditional probability

I have been working on a question for hours now and I thought I would ask stack exchange. "Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a ...
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205 views

How to understand Gibbs distribution

I have a graph model such as Following the Hammersley–Clifford theorem describes that Markov random fields exhibit a Gibbs distribution with an energy function as follows: $$P(x)=\frac ...
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1answer
36 views

Conditional probability of intersection of multiple hypergeometric distributions

What's given: I have an urn with with a set $S$ of balls where $|S| = N$. Each ball $b_i$ has a unique id and can either be red or blue. There are $m$ red balls in the urn. $d$ times I randomly draw ...