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### Independence of diagnostic tests

$P(T_1 \cap T_2 | D)$ : Probability that the results are positive in both the tests given Fred has the disease. $P(T_1 \cap T_2)$ : Probability that the results are positive in both the tests taking ...
• 26.6k
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• 40.3k
1 vote
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### Dependence of Events - Proof

If the events $A^{c}$ and $B^{c}$ are independent, then $A$ and $B$ are independent. Indeed, suppose that $A^{c}$ and $B^{c}$ are independent: \begin{align*} \mathbb{P}(A\cap B) & = 1 - \mathbb{P}(...
• 8,029
1 vote
Accepted

### What statement do I use here to get this equation $\Bbb{E}\left(S_k \cdot \Bbb{1}_{\{T=k\}}\right)=\Bbb{E}(S_k)\Bbb{E}(\Bbb{1}_{\{T=k\}})$?

You can see T as a stopping time. That is, you compute the average up to time T. The key idea, and in my opinion the easiest way to compute this, is to use the Law of Total Probability as follows,  ...
1 vote
Accepted

Suppose $\{X_i\}_{i=1}^{\infty}$ are mutually independent Poisson Point Processes (PPPs) on the real number line with intensities $\{\lambda_i\}_{i=1}^{\infty}$. For any region $B \in \mathcal{B}(\... • 20.3k 1 vote ### If$A_1,A_2,…,A_n$are independent then$A_1^c,A_2^c,…,A_n^c$are also independent. Partial attempt: Let$B_1 := \bigcap_{i=1}^k A_i^c$and$B_2 := A_{k+1}^c\$. \begin{align} P\left(\bigcap_{i=1}^{k+1} A_i^c\right) &= P(B_1 \cap B_2) \\ &= 1 - P(B_1^c \cup B_2^c) \\ &= 1 - ...
• 81.7k

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