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Probability based on a given a consecutive series
Suppose there is a series S, where each element follows a geometric distribution with parameter p > 0. The expected value of S is 2/p.
The question is what is the probability that the first 3 elements of the series exceed a given bound X?
I want to understand the intuition behind this question and not answer it by calculation.
A:
For the sake of simplicity of notation, let’s assume $X$ is positive.
Recall that the geometric distribution is defined as $\mathbb P(X = n) = (1-p)^{n-1}p$.
Let’s first bound the probability that the third element of the sequence exceeds $X$.
Since the first two elements are below $X$, the only way the third element could exceed $X$ is if the first two did not, but this happened with probability $(1-p)^2$.
Thus, the probability of the third element exceeding $X$ is $(1-p)^2(1-p)$.
Since there are $3$ elements to consider, the probability that the first three elements exceed $X$ is
$$
\mathbb P(S_3 > X) = p^2(1-p)^2(1-p) + 2p(1-p)^3 + 2p(1-p)^2 = 3p(1-p)^2(2+p).
$$
Now we just need to bound this probability for $X = n$.
The first term says that the probability that the first element exceeds $n$ is $p^2(1-p)^{n-1}$, and hence the probability that the first two elements exceed $n$ is $2p^2(1-p)^{n-2}$, and the probability that the third exceeds $n$ is $(1-p)^{n-3}$.
The second term says the probability that the second exceeds $n$ is $2p(1-p)^{n https://automative.club/upload/files/2022/06/69NXDm37g2pPwlznTylW_07_02db094ad4e4d9ac3ca8ea6f2d10f3ca_file.pdf
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The 1.3 version adds support for Windows 10 and Ultrawide. Executable, 32-bit, 64-bit, Notes. Windows . On Windows 7 and 8.1, the game is non-functional unless a keyboard or mouse is connected to the gamepad.
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Microsoft has announced that the August 2004 Update for Windows 10 will also disable the graphics acceleration for non-SMP and non-SLI configurations. As a result of that change, multi-GPU configurations are becoming obsolete. The Xbox 360 S and PlayStation 3 Slim both use NVIDIA’s GTX 200 series GPUs, while the Xbox One uses AMD’s Polaris 12 GPU.
Logo
The official logo of the Resident Evil franchise is a swirl of red and black with the letters “R.E.” written in white. It was used by Capcom and its licensees. The official logos are the same for every game that has been released in the series.
In the Resident Evil series, the logo itself has had several designs over the years. One of the more notable designs is featured in Resident Evil.
See also
List of PC games released in 2004
List of weapons in Resident Evil
Notes
References
External links
Resident Evil at the Internet Movie Database
Category:Action-adventure games
Category:Capcom franchises
Category:Video games about the United States Marine Corps
Category:Video game franchises introduced in 1998
Category:Video games about the United States Marine Corps
Category:Video games developed in Japan
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