All Questions
Tagged with mathematical-statistics conditional-probability
154
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Formal definition of sufficient statistic
Let $(\Omega_X,\mathcal{F}_X)$ and $(\Omega _T,\mathcal{F}_T)$ be measurable spaces. Let $\mathfrak{M}$ be a family of probability measures on $(\Omega_X,\mathcal{F}_X)$. Let $X:\Omega\to \Omega _X$ ...
- 133
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115
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Borel-Cantelli lemma on conditional probabilities
In a probability space $\big( \Omega, \mathcal{F}, P \big)$, suppose $\{E_n\}_{n\in \mathbb{N}} \subseteq \mathcal{F}$ is a sequence of mutually independent events. By Borel-Cantelli Lemma, the ...
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24
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The concept and notation about MLE(Likelihood) and MAP
In generally, we say that X1, X2, ..., Xi are from a certain distribution, which can be represented by f(x;θ), where θ is an unknown parameter.
When I read content related to MLE or the Likelihood ...
- 101
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58
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Expected value of a variable that depends on two other random variables, and one of these random variables depends on another random variable
I am trying to solve the problem using conditional expectations. The expected value H is depends on the waiting time T and a set threshold X (a real number that is a constant random variable during ...
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If $P(A|D) > P(A)$ and $P(B|D) > P(B)$, then is $P(A \cap B|D) > P(A \cap B)$?
There are 3 events $A, B, D$ such that $D$ makes $A$ more likely and $D$ makes $B$ more likely. Does this mean that $D$ makes it more likely that both $A$ and $B$ occur? How can you prove this using ...
- 11
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46
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How to mathematically express the fact that the conditional probability $P(Y|X)$ can be independent of $P(X)$?
Mathematically, $P(Y|X) = \frac{P(X,Y)}{P(X)}$ and so $P(Y|X)$ must depend on $P(X)$. Since $P(Y|X)$ will change when $P(X)$ changes.
However, consider this scenario:
X = amount of red meat consumed ...
- 217
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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} ...
- 51
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58
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Binomial distribution conditional on the weigthed sum?
Suppose $\mathbf{X}$ is a vector of iid Bernoulli variables with the fixed success probability of $p$. The variance of X is $np(1-p)$.
Now, suppose, I am interested in the conditional probability of $...
- 19
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16
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Expression for Markov Kernel sampling indeces in $\{0, \ldots, T\}$ according to weights depending on another variable
I have a vector $x = (x_0, \ldots, x_T)$ and given this vector, I would like to sample an index $k$ between $0$ and $T$. The probability of sampling index $k$ is given by a weight $w_k$ that is a ...
- 2,236
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Formula of $\text{Var}[X|Y,Z]$ for $X\sim \mathcal N(\mu_X,\sigma_X^2)$, $Y\sim \mathcal N(\mu_Y,\sigma_Y^2)$, $Z\sim \mathcal N(\mu_Z,\sigma_Z^2)$? [duplicate]
How do I condition the variance of a normally distributed random variable on two other normally distributed random variables?
How do I condition the expectation of a normally distributed random ...
- 47
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447
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Prove that the sum is sufficient using using the definition of sufficiency
If $X_1,\ldots,X_n$ is an IID random sample, with $X_i\sim\,\text{Ber}(\theta)$, prove that $Y = \sum_i X_i$ is sufficient using the definition of sufficiency (not the factorization criterion).
Now ...
- 145
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644
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Ising's model and conditional probabilities
I am trying to understand how the conditional probabilities of an Ising model having the following joint probability is given by a logistic regression model, as shown in the joint image. I'm more than ...
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2
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203
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How to find the probability of an unobserved binary variable from repeated noisy observations?
Let $Y \in \{0,1\}; P(Y=1)=\beta$. We have no observations of $Y$.
Instead, we observe a sample of $A$,$B$. We can assume that $P(A,B|Y)=P(A|Y)P(B|Y)$; $P(A=Y)=P(B=Y)=\alpha$; and that $P(A=Y|Y)=P(A=Y)...
- 181
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32
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Estimating probability of an event at multiple points in time
I have Machine learning Model which estimates the churn probability in the next 6 months , for a given customer.
Is there any mathematical way to estimate the ...
- 141
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93
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hypothetical statistical test - type I and type II errors
A hypothetical statistical hypothesis test that can be used for any type of hypothesis is conducted by drawing a random number between 0 and 1 and rejecting the null hypothesis if it is less than 0.05,...
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