Questions tagged [conditional-probability]
For questions on conditional probability.
5,998 questions
3 votes
1 answer
52 views
Conditional probability for linear combinations of independent exponentials
I am working on the following exercise. Let $$X_1 \sim \mathrm{Exp}\left(\tfrac12\right), \qquad X_2 \sim \mathrm{Exp}\left(\tfrac12\right),$$ independent. Define $$Y_1 = X_1 + 2X_2, \qquad Y_2 = 2X_1 ...
- 375
4 votes
3 answers
284 views
Prove a formula on conditional probability: $E(g(X,Y)| Y=y)= E(g(X,y)|Y=y)$.
Let $(\Omega,\mathcal F, P)$ be a probability space and $Z:\Omega\to\mathbb R$ be a random variable. $Y: (\Omega,\mathcal F)\to (\Lambda,\mathcal G)$ is a measurable map between these two measurable ...
- 2,850
1 vote
1 answer
100 views
The number of positive outcomes is independent of the number of negative outcomes (under Poisson)
I conduct $X \sim \text{Poisson}(\lambda = 1)$ experiments. Each experiment is IID, with probability $p$ of outcome $\bf A$ and $q = 1-p$ of $\bf B$. Let $A, B$ be the total number of experiments with ...
- 6,361
-1 votes
2 answers
39 views
Conditioning a discrete random variable on a continuous random variable [closed]
Let $X$ be a discrete random variable and $Y$ be a continuous random variable. I can't seem to come up with an explanation as to why $X|Y$ is always discrete. How do I see this intuitively?
- 117
2 votes
0 answers
57 views
Tower Property implications on the conditional distribution
The tower property (law of total expectation) states that for any $σ$-subalgebras $G_1 ⊆ G_2$ $$ \text{(I)} \qquad E[X∣G_1] = E[E[X∣G_2] ∣ G_1] \qquad\text{a.s.} $$ In particular, for an integrable ...
- 12.3k
2 votes
0 answers
83 views
Convergence of Joint Distributions with Conditional Independence: $(X_n, Z_n) \to (X, Z)$?
Suppose that you have sequences of three random variables $X_n, Y_n, Z_n$ which converge in distribution to rvs $X, Y, Z$. Suppose that the distribution of $(X_n, Y_n)$ converges uniformly to the ...
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1 vote
1 answer
132 views
Conditional probabilities in a sudoku
While making sudoku puzzles I came up with the following question: Suppose there is a square $a$ in which there can be a $3$ or a $4$. Obviously, the probability of there being a $3$ is equal to that ...
- 353
0 votes
2 answers
83 views
Independence of Experiments in Conditional Coin Toss Experiments
Consider the following experiment: toss a fair coin until the first head appears. The sample space is: $S=\{H_1,T_1H_2,T_1T_2H_3,T_1T_2T_3H_4,…\}$ Now, take two events: $E_1=\{T_1H_2\}$ and $E_2=\{...
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