5 votes

Uniform Distribution of Sequences Modulo $2\pi$

It's more traditional to study equidistribution mod $1$ rather than $2\pi$, but of course they are related: $a_n \mod 1 = (2 \pi a_n \mod 2\pi)/(2\pi)$.See Equidistributed sequence. Equidistribution ...
Robert Israel's user avatar
3 votes
Accepted

The discrete PDF $p_\alpha(k) = |\binom{\alpha}{k}|$, $\alpha \in (0,1)$

It is called the Sibuya distribution with generating function $1-(1-z)^{\alpha}.$ Just google for numerous references.
Letac Gérard's user avatar
2 votes

Find CDF of U = X/Y

$(X,Y,1-X-Y)\sim \frac{(Y_1,Y_2,Y_0)}{Y_1+Y_2+Y_3}$ (where $Y_1,Y_2,Y_3$ are exponential) is Dirichlet $(1,1,1)$ distributed. Therefore $$\Pr(X>qY)=\Pr(Y_1>qY_2)=E(\Pr(Y_1>qY_2|Y_2))=E(e^{-...
Letac Gérard's user avatar
2 votes
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How to Determine Independence of Events Using Probability

Independent has a very specific meaning in probability, namely events $A$ and $B$ are independent if and only if $Pr(A|B)=Pr(A)$. A consequence of this, which comes from the rule $Pr(A|B)=\frac{Pr(A\...
Red Five's user avatar
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2 votes

$E[X|X^2+Y^2] = 0$ when $X$ and $Y$ are independent standard normals.

Let $A$ be a Borel set in $\mathbb R$. Then, $\int_ {(X^{2}+Y^{2}\in A)}XdP=\int_ {(x^{2}+y^{2}\in A)}xdF_{X,Y} (x,y)$ and $\int_ {(X^{2}+Y^{2}\in A)}(-X)dP=\int_ {(x^{2}+y^{2}\in A)}xdF_{-X,Y} (x,y)$....
geetha290krm's user avatar
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2 votes

Uniform Distribution of Sequences Modulo $2\pi$

The standard method for proving equidistribution is Weyl's criterion which states that a sequence $a_r$ is equiditributed modulo $1$ if and only if for all non-zero integers $k$, $$ \lim_{n \to \...
Nilotpal Sinha's user avatar
2 votes
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Strange consequence of linear combination of normal distribution

You have added "independent" to your claim. But then the $Y+Y$ in your question is confusing. If $Y_1 \sim N(0,1)$ and independently $Y_2 \sim N(0,1)$ then $U=Y_1+Y_1=2Y_1 \sim N(0,4)$ ...
Henry's user avatar
  • 157k
2 votes
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Random variable obtained from discrete and continuous Random variables

The hierarchical model is $$X \sim \operatorname{Uniform}(0,1) \\ Y \mid X \sim \operatorname{Bernoulli}(X) \\$$ and $Z = (X \mid Y = 1)$. So it is natural to first compute the distribution of $X \...
heropup's user avatar
  • 137k
2 votes

Compare two distributions

KL divergence, although neither a metric nor symmetric, is the most used measure used to measure distance between two probability distributions. Check out more, here: https://en.wikipedia.org/wiki/...
whoisit's user avatar
  • 3,179
2 votes
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Convergence in Probability how to solve

You want to show that $\lim_{m\rightarrow\infty}P(|X_{m} - X| > \epsilon) = 0$, where $X$ is the zero random variable. Given that $X_{m}$ can take positive value $m^2$ with probability $\frac{1}{m}$...
rcescon's user avatar
  • 241
1 vote

Choosing Null Hypothesis

I disagree with the choice $H_A : \mu < 500$. The wording of the question suggests that the company wishes to test the hypothesis that the mean lifetime is at least $500$ hours. This means that ...
heropup's user avatar
  • 137k
1 vote

What confluent hypergeometric function equality is needed for this integral?

The essential step is to isolate the $\theta$-dependent term, substitute $ \cos \theta = z,\ d\theta=-\frac{dz}{\sqrt(1-z^2)} $ such that a relatively simple integral results $$d\theta \cos (\theta ) ...
Roland F's user avatar
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1 vote

If $X$ is sub-Gaussian random variable with variance proxy $\sigma^2$, how to show that $E\{ \exp( t X^2) \} \leq (1 - 2 t \sigma^2)^{-1/2}$?

You have just to use the integral representation $$e^{tX^2}=\int_{-\infty}^{\infty}e^{u\sqrt{2t}X-\frac{u^2}{2}}\frac{du}{\sqrt{2\pi}}$$ and Fubini. Edit: Oh, I missed the point: what to do if $t<...
Letac Gérard's user avatar
1 vote

Random variable obtained from discrete and continuous Random variables

You are looking for the pdf for $Z$ , not its CDF. You have the pdf for $X$, and the conditional pmf for $Y$ given $X$; since you know their distributions. $\begin{align}f_X(x) &= \mathbf 1_{0\leq ...
Graham Kemp's user avatar
1 vote

Why does it look like the probability exceeding 1? How do you solve this problem?

To summarize the discussion in the comments: We are after the expected probability of failure, and we compute that as $$\int_0^1 x\times 2(1-x)\,dx=\frac 13$$ Thus you expect the, randomly selected, ...
lulu's user avatar
  • 70.6k
1 vote
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Calculating Probability in Disjoint and Independent Events

No, it's not true that $\ P(A'\cap C'\,|\,B)=\frac{P(A'\cap\,C')}{P(B)}\ .$ By definition $$ P(A'\cap C'\,|B)=\frac{P((A'\cap C')\color{red}{\cap B})}{P(B)}\ ,\tag{1}\label{e1} $$ and since \begin{...
lonza leggiera's user avatar
1 vote

Robot Capture the flag

I think the initial strategy from the other solution is correct, but it's possible to update Aaron's strategy to improve his probability of winning somewhat. I'm not completely sure that my ...
math kuma's user avatar
1 vote

Robot Capture the flag

I feel like I may have misunderstood the question so please tell me if I have done so. Firstly, it is reasonable to find what the optimal strategies actually are for Erin and Aaron. Since Erin knows ...
Sai Mehta's user avatar
  • 1,161

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