New answers tagged probability
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Probability of sum and conditional expectation
Note $(X,Y) \sim \mathrm{Pois}(\lambda)\otimes \mathrm{Pois}(\mu)$. Using generating functions (probability generating functions, moment generating function, or characteristic function all work here), ...
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Accepted
A negative Kullback-Leibler divergence between two Laplace distributions?
The formula you got is correct for $\log$ in base $e$ (natural log). Sadly, Desmos understands that $\log = \log_{10}$. Replace that by $\ln$ and you'll get the right graph.
Besides (using natural log)...
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Proof about Markov kernels and absolute continuity
Assuming the following conditions:
$(\mathsf{X}, \mathcal{X})$ is a measurable space.
$M_n$ and $L_{n-1}$ are Markov probability kernels for $n=2, \ldots, P$.
$\mu_n$ are probability measures on $(\...
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Prove that for an increasing series of events, Event N is equal to the union of events 1 to N.
If $x$ is in your union, by definition $x$ belongs to some $E_k$ with $k\le n$. But any such $E_k$ is a subset of $E_n$. That is all there is to it.
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The Probability of Two Contestants Meeting (Ross)
It seems easier to compute $\Pr[E]$ unconditionally. There are $2^n-1$ total games played, out of $\binom{2^n}{2} = 2^{n-1}(2^n-1)$ possible games; by symmetry, each possible game has a $\frac1{2^{n-1}...
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If the sides of triangle are decided by throwing a die thrice the probability that the triangle is a equilateral triangle or isosceles triangle is
There are the following possibilities to construct the triangles:
$$
\begin{array}{llll}
a& b& c & \#\\
x&x&x& 6\\
2&2&1,3 & 2\times 3=6\\
3&3& 1,2,4,5 &...
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If the sides of triangle are decided by throwing a die thrice the probability that the triangle is a equilateral triangle or isosceles triangle is
According to N.F. Taussig's comment: ...the requirement for a triangle is that the sum of the lengths of any two sides must exceed the length of the third side. So it's not about the squares.
Also, by ...
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$X$ is sub-Gaussian, then $X^2$ is sub-exponential
I cannot find a proof, but with the extra hypothesis $E(X^2)=\sigma^2$ everything is easy. If $2s\sigma^2<1$ we can write
$$E(e^{tX^2})=E\left(\int_{-\infty}^{\infty}e^{sX-\frac{s^2}{2t}}\frac{1}{\...
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An apparently easy game of chance
Let $X$ be the amount of rounds at the end of the game.
Then $$P(X\le x) = \left(1- a^x\right)^n$$
for $a=\frac12$ (this is the same approach as @lulu's comment). Hence
$$P(X>x)= 1-P(X\le x) = \...
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Biased probability
A game ends in a given round when it is not the case that all three flipped the same result. This happens with probability $1-\underbrace{0.9\cdot 0.6\cdot 0.4}_{HHH} - \underbrace{0.1\cdot 0.4\cdot ...
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Biased probability
Let P(T) be the probability that Terry wins on a given round. For Terry to win, he must flip heads while both Tom and Jay flip tails.
The probability of Terry flipping heads is 0.9, and the ...
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Proof of Conditional Expectation using Indicator Random Variable
The line $X = \sum_{k = 0}^\infty k \cdot \mathbb{I}_{X = k}$ makes use of the fact that the indicator variable $\mathbb{I}_{X = k}$ is defined to equal 1 when $X = k$, and 0 otherwise. So as long as $...
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Problem of the urn and hypergeometric distribtuion
Classically, probability is the number of successful outcomes, divided by the number of total outcomes.
For your problem, both of these numbers are binomial coefficients.
In particular, the total ...
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Probability (expectation)
Each type has an equal probability of being picked/not picked in each draw, thus
P(type i not picked in a draw) $=\frac{19}{20}=0.95$
so P( type i not picked in $10$ draws) $= 0.95^{10}$
and P(type i ...
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Simultaneous birthday probability. Why is there an $(365-n+1)$?
Look at it as an arithmetic sequence.
$365, 364, 363, ...$
The first term $a = 365$
The common difference $d = -1$
The $n^{th} \ term = a + (n - 1)d = 365 + (n - 1)(-1) = 365 - n + 1$
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Why does the Elo rating system work?
I didn't see anyone mention--though it is possible Glickman mentions it in the linked articles--the Bradley-Terry model from which the formula for expected score based on ratings is derived: https://...
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Probability (expectation)
Blockquote
$20^{10}$ = The number of different ways 10 coupons can be selected from 20 types, assuming repetition is allowed.
The number of ways all 10 coupons are same = $^{20}C_1 \times ^{19} C_0$
...
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How to show the convergence of a random variable does not imply convergence of expectation
Isn't this just by showing that the expectation of the limit is just 1, and that the limit of the expectation is 0?
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Probability of $x$ trials given $k$ successes
Your question is relevant to the concept of conditional probability. Recall the Bayes' theorem
$$P(X=x|k) = \frac{P(X=x)P(k|X=x)}{P(k)}=\frac{P(X=x)P(k|X=x)}{\sum_i P(X=i)P(k|X=i)}$$
We can back out ...
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Problem about Uniform distribution
We may approach this problem by considering the number of arrangements that fulfil the requirement.
For $n$ people sitting at a roundtable, the total number of permutation is $(n-1)!$.
Therefore, the ...
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Proving probability of intersection greater than or equal to probabilities of events - N + 1
$$
\begin{align*}
P \left( \bigcap_{j = 1}^N A_j \right) &= 1 - P \left( \bigcup_{j = 1}^N A_j^\complement \right) \geq 1 - \sum_{j = 1}^N P\left( A_j^\complement \right) \\
&= 1 - \left( \...
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Expected Maximum Value of 10 Randomly Selected Balls from an Urn
There is a clever method using linearity of expectation. I got this argument from two of joriki's answers: one about cards in a deck and one about the continuous version of points on a line.
First, ...
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Expected Maximum Value of 10 Randomly Selected Balls from an Urn
You can use binomial coefficient identities to simplify the fraction:
$$
\frac{\sum_{k=10}^{20} k \binom{k-1}{9}}{\binom{20}{10}}
= \frac{\sum_{k=10}^{20} 10 \binom{k}{10}}{\binom{20}{10}}
= \frac{10 \...
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$E(Z_iZ_j) = E(Z_i)E(Z_j)$ for independent r.v.s
First, it is confusing to say that "$X$ is a random variable". In the following, I interpret that, for each individual index $k\in\{1,\dots,N\}$, the quantity $X_k$ is already fixed. I ...
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Expected Value of $X^4$ if $X$ ~$N(0, \sigma^2 )$
Differentiating w.r.t. $\mu$ any number of times preserves the exponential factor, so that the integral can be expressed as a linear combination of expectations:
$$\begin{align*}
\frac{\partial^2}{\...
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Accepted
Reconciling Definitions of Conditional Expectation
In the second definition take $\mathcal G=\sigma(Z)=\{Z^{-1}(E): E \,\, \text {Borel in } \mathbb R\}$. If the equation in the first definition holds, you can take $h=1_E$ to get the equation in the ...
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Show $P_n = \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \dots + \frac{(-1)^n}{n!}$ given $P_n = \frac{n-1}{n} P_{n-1} + \frac1n P_{n-2}$
See given that,
$$ P_n = \sum_{k=2}^n \frac{(-1)^k}{k!} \tag{1} $$
we want to re-write $(1)$ in terms of $P_{n-1}$ and $P_{n-2}$ so,
$$ P_n = \sum_{k=2}^{n-1}\frac{(-1)^k}{k!} + \frac{(-1)^n}{n!} $$
$...
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Accepted
Show $P_n = \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \dots + \frac{(-1)^n}{n!}$ given $P_n = \frac{n-1}{n} P_{n-1} + \frac1n P_{n-2}$
$$P_n = \frac{n-1}{n} P_{n-1} + \frac1n P_{n-2}
\\ P_n - P_{n-1} = -\frac{1}{n}\left({P_{n-1}-P_{n-2}}\right)
$$
Let : $V_n=P_n - P_{n-1}$
Therfore:
$$V_n=-\frac{1}{n}V_{n-1} \implies V_n=(-1)^n\frac{...
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Checking consistency of minimizer given uniform convergence of its objective function
If I understand your question correctly, then, $a_n^2\to 0$, thus $O_p(a^2_n)\to 0$ in probability. Also, $\hat{\theta}_1 \to \theta_0$. Therefore $\hat{\theta}_2 \to \theta_0$. Let me know if this is ...
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Dice conditional expectation
Found the trick to solve it. Let $n$ be the max value of the die, and $(d_1,d_2)$ be a pair satisfying $d_2 > d_1$, ie. a favorable event, where the second element of the pair represents your roll, ...
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The matching problem in a recursive view
The probability that no one sits at the right seat will be that $1$ person will sit in the $n-1$ places that do not belong to them, and the next person will sit in the $n-1-1=n-2$ that do not belong ...
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Expected Maximum Value of 10 Randomly Selected Balls from an Urn
Distribute the $20$ numbers in ascending order uniformly on a $0-1$ scale, so they're at $\frac{k}{21},$ for $k=1,2,3,...20$
On the other hand, in similar vein, the sampled numbers are at $\frac1{11},...
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The matching problem in a recursive view
Instead of considering full table try to consider emty table.
We have two cases.
First guest seats on place numbered $i$, next comes guest numbered $i$. He can seat on first place or other. If seats ...
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Why Do Fewer Points Result in Larger Variances?
There are a lot of unstated assumptions in your question, which is what is leading to your confusion.
First of all, what does it mean to say "the variance of these data points?" One ...
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Why Do Fewer Points Result in Larger Variances?
The Crux of this Query is the confusion over "variance" & "variance of variance" !
At the end of the Post , I will re-write what OP has written , to make it Accurate & ...
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Accepted
Why is a sequence of random variables not a markov chain?
Let's analyze the given information and the problem. The terms $Y_1$ and $Y_2$ are i.i.d. Bernoulli$(0.5)$, and for $j \geq 3$:
if $\min(Y_{i-1}, Y_{i-2}) = 1$, then $Y_i$ is Bernoulli$\left(\frac{2}{...
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Where am I going wrong with evaluating this integral?
By integration by parts and then L’Hospital Rule
\begin{aligned}
\int_{200}^{\infty}\left(-x e^{-x / 1000}-1000 e^{-x / 1000}\right) d x
= & 1000 \int_{200}^{\infty}(x+1000) d\left(e^{-x / 1000}\...
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Show $1/X_t^2$ is bounded where $X_t$ is a random process
I'm going to try and provide a very explicit answer to your question. The main trick is to give an upper bound expression that is in general much easier to work with.
To start let $t \in \mathbb{N}$ ...
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Maximize product of die
Your work is wrong ($D1$ and $D2$ are not independent) and your script is probably correct. The actual value is $\frac{617}{36} \approx 17.13889$.
To calculate this, consider the expected product for ...
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Proving that a function is negligible
Let me only draw a little picture over here for the case we pick $c=2$, to help a little bit the intuition.
In addition, as John Bentin wrote, let me highlight that we are not obliged to find the ...
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Proof that if $Z = X + Y$, where $X\text{~Bin}(n, p)$ and $Y\text{~Bin}(m, p)$ and are independent, then $Z\text{~Bin}(n + m, p)$.
You may also apply the Law of Total Probability and the independence of $X$ and $Y$:
\begin{align*}
\mathbb{P}(Z = z) & = \mathbb{P}(X + Y = z)\\
& = \sum_{x = 0}^{\infty}\mathbb{P}(X = x, Y = ...
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Proof that if $Z = X + Y$, where $X\text{~Bin}(n, p)$ and $Y\text{~Bin}(m, p)$ and are independent, then $Z\text{~Bin}(n + m, p)$.
This is not sufficient; it only shows that the means match. But distributions are much more than their means.
What you need to argue is simple: making $x$ binomial($p$) decisions "followed by&...
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Explain $E=\frac{1}{2}\left(E+\frac{2}{3}\right).$
Either you get to roll (with P = 3/6) or you dont.
In case you do, you'll earn E more with
P = 3/6 of earning 1 more (rolling 1 or 2 or 3)
P = 2/6 of earning 0 (rolling 4 or 5)
P = 1/6 of losing 1 (...
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Accepted
Given two continuous random variables X and Y with different domains, how can you calculate P(X<Y) given some joint PDF?
When you are finding the bounds for integration for computing $P(X<Y)$, you need to take all inequalities into account. We are given that $0\le x\le 20$ and $0\le y\le 400$. You want to find $P(X&...
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$Cov(X_t , X_{t-2})$ in $AR$ model.
Let us work with the AR(1) without the constant term $Y_t = X_t-\mu$ which is more direct. As mentioned by @Henry, you can write
$$
\begin{aligned}
Y_t &= aY_{t-1} + Z_t \\
&=a(aY_{t-2}+Z_{t-1}...
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Find the probability that at least 2 defective bulbs are drawn, if 4 bulbs are drawn from a box containing 10 bulbs of which 3 are defective.
$$\frac{\binom{3}{2}\times \binom{7}{2}+ \binom{3}{3}\times \binom{7}{1}}{\binom{10}{4}}=\frac{1}{3}$$
Number of ways choosing $2$ defective out of $3$ and $2$ nondefective out of $7$ is $\binom{3}{2}\...
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Shannon entropy property
Im not sure how to use the given inequality here, but one way to shoe the claim is by Jensens inequality. Notice that:
$$H(x) = -\sum x_i \log x_i$$
Is concave with respect to $x$. Rewriting your ...
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The matching problem - placing $n$ letters into $n$ envelopes randomly - the meaning of the sum of $\operatorname{Pr}(A_i)$.
This is an example of linearity of expectation. For a Bernoulli random variable $X$, we have $$E[X]=0\cdot P[X=0]+1\cdot P[X=1]=P[X=1].$$ Let Bernoulli random variable $X_i$ indicate whether letter $...
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Probability of card game
To make things easier, always draw 10 cards regardless if we win or not. We use inclusion-exclusion principle over the events $X_1,X_2,\dots X_{10}$ where $X_i$ is the event that position $i$ has a ...
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Accepted
Probability of getting a satisfactory grade on a test.
There are actually only $5$ cases to consider. Let $S$ be the event that the first student gets a satisfactory grade, and $A$ be the number of problems he solved by himself originally.
For (1), find $...
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