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Conditional PDF of bivariate normal

You have learnt in class that if $(Z_1,Z_2)\in R^n\times R^d$ is Gaussian with covariance by blocks $$\Sigma=\left[\begin{array}{cc}A&B^T\\B&C\end{array}\right]$$ then the mean of $Z_1|Z_2$ is ...
Letac Gérard's user avatar
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How to create a smooth distribution of data with specific n, min, max, mean?

I figured out a way to do this using LOGNORM.DIST() function in Excel. I believe the same method can be done with NORM.DIST() as well. This method requires trial and error. Below are two images for ...
Aw CheeHong's user avatar
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Normal distribution and conditional expectation

Okay, it turns out it’s just a simple integral calculation, you don’t even need to rewrite it as a conditional expectation. Consider the following function: $$ q(a)=\int_{a}^{\infty}x f(x)dx $$ where $...
Egor Larionov's user avatar
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
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What is the probability that one simultaneous roll of five dice gets a four?

The event whose probability you want to find is apparently a four-of-a-kind, four dice showing the same number and one showing a different number. The proposed solution in the question is $$ \frac{\...
David K's user avatar
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A box contains many balls where $25$% are red. $1200$ balls are selected at random

Your answer to part (a) is correct. For part (b), you have the right idea, but for best results, you will want to employ continuity correction. If $X$ is the random number of balls in the sample that ...
heropup's user avatar
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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
3 votes
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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
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What is the probability that one simultaneous roll of five dice gets a four?

You have stated the question rather obscurely, but on careful reading, I believe that you are asking for one face to show $4$ times, and another different face to show $1$ time on the $5$ dice thrown, ...
true blue anil's user avatar
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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Distribution of sample variance of Cauchy distributed variables

FWIW I can offer a straightforward explicit result for the case of $n=2$. Call the sample variance $S$, and let the $x_i$ be standard Cauchy random variables. We have the purely algebraical elementary ...
lupus's user avatar
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Robot Capture the flag

I'm not sure why the other answer was not correct, but an alternative approach I was thinking of was to take Erin's optimal position as being at $x = \frac{1}{2}$, while Aaron's optimal position is at ...
math kuma's user avatar
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Random variable obtained from discrete and continuous Random variables

$(X_n,Y_n)$ $n=1,2,\ldots $ are iid. If $T=\inf\{n; Y_n=1\}$ we are asked for $\Pr(X_T<z).$ $$\Pr(X_T<z, T=n|X_1,\ldots,X_{n-1},Y_1,\ldots,Y_{n-1})=\Pr(X_n<z \cap \{Y_n=1\})(1-X_1)\ldots (1-...
Letac Gérard's user avatar
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
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
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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
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Memoryless property of the geometric distribution

Here is a detailed, step by step proof. $$\begin{align} \Pr[X > s + t \mid X > s] &= \frac{\Pr[(X > s + t) \cap (X > s)]}{\Pr[X > s]} \\ &= \frac{\Pr[X > s + t]}{\Pr[X > s]...
heropup's user avatar
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0 votes
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Form of joint distribution of Markov model

$\def\iddots{\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.}}\\ \def\A{\mathbf{A}}\\ \def\x{\vec{x}}\\ \def\w{\vec{w}}\\ \def\Q{\mathbf{Q}}\\ \def\0{\vec{0}}\\ \def\I{\mathbf{I}}\\ \def\...
lonza leggiera'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
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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
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Strange consequence of linear combination of normal distribution

The result is true only when the random variables are independent.The proof is simple. Without the assumption of independence, it is clearly false. A simple counter-example is $U = Y-Y = 0$ which is ...
Ibra's user avatar
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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
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
0 votes

If $X \sim N(μ,σ^2)$ how to find the PDF of $Y = 2X$

I know this question has been answered already, but out of curiosity I thought it was a good idea to tell you that you can find the distribution of $Y$ with the non-central moment generating function ...
Tiago Coelho'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
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0 votes

Question about the null fraction parametrization for a poisson model (Mathematical statistics)

$\eta=P_{\lambda}(X=0)$ which is a probability in $[0,1]$. Meanwhile $\eta=0$ will not give a (Poisson) distribution and is equivalent to $\lambda=+\infty$, while $\eta=1$ will give a distribution ...
Henry's user avatar
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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

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
  • 3,169
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
Accepted

Does the kurtosis need to be finite for the sample variance to be consistent?

We do not need the Kurtosis to be finite for so called weak consistency (convergence in probability), but the so called strong consistency (convergence in the 2nd moment) is equivalent to that the ...
Amir's user avatar
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1 vote
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Find CDF of U = X/Y

For each real number alpha, let's compute the probability that $X/Y < \alpha$. If $\alpha \leq 0$ this probability is obviously $0$. Otherwise, it is the probability that $X < \alpha Y$, which ...
hunter's user avatar
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1 vote
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Expected value of the random variable: $\int (\mu + \sigma Z)\phi(x)dZ$

$$ \begin{split} \mathbb{E}[\mu + \sigma Z] &= \int_{-\mu / \sigma}^{\infty} (\mu + \sigma z) \phi(z)dz \\ &= \mu \int_{-\mu / \sigma}^{\infty} \phi(z)dz + \sigma \int_{-\mu / \sigma}^{\infty} ...
finch's user avatar
  • 1,646
1 vote
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Compare two distributions

Because you appear to be looking for authoritative references discussing distances between probability distributions, here is a non-comprehensive list of references. (Making this community wiki so ...
2 votes
Accepted

PDF of $Y=g(X)$ when $X\sim N(0,1)$. $g(X)$ is a piecewise function where each part is constant.

Your expression for the CDF of $Y$ is close, but should be $$F_{Y}\left(y\right)=\begin{cases} 0, & y<-1\\ 1/2, & -1\leq y<1\\ 1, & y\geq1 \end{cases}.$$ Since $Y$ is discrete, it ...
AOS's user avatar
  • 181
0 votes

Integral of Cauchy distribution?

I think the answer is wrong. There is no way that two different distributions share the same CDF. Moreover, the answer mentions $\mu_1, {\sigma_1}^2$, but the question only has $a_i,b$. To be specific,...
Y.D.X.'s user avatar
  • 364
0 votes

Suppose $Y\backsim N(0, 1)$ and $X\backsim N(0, Y^{-2})$. Show that $X$ has the standard Cauchy distribution.

$X\sim N(0,1/Y^2)$ implies $X\sim Z/|Y|$ where $Z\sim N(0,1)$ is independent of $Y$. Therefore using $$\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}=\int_0^{\infty}\frac{u^{a-1}}{(1+u)^{a+b}},\ \Gamma(p)\...
Letac Gérard's user avatar
1 vote

Suppose $Y\backsim N(0, 1)$ and $X\backsim N(0, Y^{-2})$. Show that $X$ has the standard Cauchy distribution.

If $$X \mid Y \sim \operatorname{Normal}(0, Y^{-2}),$$ then the variance of $X \mid Y$ is $1/Y^2 > 0$, and the conditional density of $X \mid Y$ is $$f_{X \mid Y}(s \mid t) = \frac{|t|}{\sqrt{2\pi}}...
heropup's user avatar
  • 137k
0 votes

Help me intuit Equal Probabilities in Poisson Distribution for $k = λ$ and $k = λ-1$

For the Poisson distribution only if $\lambda$ is an integer, we have $$\mathbb P (X=\lambda)= \mathbb P (X=\lambda-1).$$ In fact, these are both the modes of the distribution. If $\lambda$ is NOT ...
Amir's user avatar
  • 5,144
2 votes

$\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$, upper bound second moment from first moment

Despite your "It's easy to show ...", Bernoulli random variables with $\mathbb P(X=1)=p$ are a counter-example: $$\dfrac{\mathbb{E}[X^2]}{ \mathbb{E}[X]^2} = \dfrac{p}{p^2} = \dfrac1p$$ and ...
Henry's user avatar
  • 157k
6 votes
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Find the pdf of $Y=\frac{X}{X+1}$

The first step is to think about the support of $Y$. If $X \in [0,1]$, then what is the range of $Y = f(X) = X/(X+1)$? When $X = 0$, we have $Y = 0$. But when $X = 1$, then $Y = 1/2$. Intuition ...
heropup's user avatar
  • 137k
2 votes
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Rearranging exponentially distributed random variables by size

There is an elegant proof using thinning/merging of Poisson processes (and exponential racing that stems from those concepts), but I am not sure if OP is aware of this concept. So, let me tackle this ...
Sangchul Lee's user avatar
2 votes

Rearranging exponentially distributed random variables by size

Let $X_1,\dotsc, X_n$ be i.i.d exponetial random variables with rate $1$ and let $X\sim \text{Exp}(1)$. Define the random variables $$ Y_{i} = X_{(i)} - X_{(i-1)} \quad (1\leq i \leq n) $$ where $X_{(...
Sri-Amirthan Theivendran's user avatar
2 votes

What is the formula that connects the average distance to the nearest point and the average number of points per unit volume?

Your formula is pretty good approximation. Even better approximation can be achieved in the following way. Assume that distribution of points is uniform and uncorrelated, so that it obeys the Poisson ...
user's user avatar
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1 vote
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Hypothesis Test using Confidence Intervals

You need to form the interval from the sample mean, not the total number of heads: $\hat{p} = \frac{9207}{17950} \approx 0.51 \implies \hat{\sigma}^2=\hat{p}(1-\hat{p}) \approx 0.25 \implies \frac{\...
Annika's user avatar
  • 6,908
0 votes

Does covariances uniquely determine the joint distribution?

For constructing $X\sim N(0,\Sigma)$ you write $\Sigma =UD^2U^T$ with $D=\mathrm {diag}(d_1,\ldots,d_n)$ and $U$ orthogonal matrix. Next you take $Z=(Z_1,\ldots,Z_n)$ with the $Z_i$'s independent and $...
Letac Gérard's user avatar
0 votes

Radon-Nikodym Derivative of a Mixed Distribution

The probability measure of an $X$ with the CDF you illustrated can be written as $\mu_X(A) = \mu_D(A) + \mu_C(A)$ where the discrete part is written $\mu_D(A) = \frac{1}{2} I_{0 \in A}$ where $I_{0 \...
Guillaume F.'s user avatar
1 vote

Probability of $k$ successes in $n$ trials where chance of success is $\frac1x$

Not really an answer. We want the distribution of the random variable $$Y = \sum_{i=0}^{n-1} X_i$$ where $X_i$ are independent Bernoulli variables with $p_i = P(X_i=1)=\frac{1}{x+i}$. This is a ...
leonbloy's user avatar
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1 vote
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Uniform distribution problem solving

Consider one end (left end which I'll consider to be the back end) of the car to be at $x$ meters away from $0$. if say $3<x<5$, then the remaining space is $x$ meters behind the car(which is ...
Mr.Gandalf Sauron's user avatar

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