Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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3
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1answer
183 views

Pushforward Measure of $\frac{x}{x^2+1}$

I was doing some exercices that was offered during the lecture and came across very interesting one. I didn't make it, then asked my teacher and didn't understand what He tried to explain me. So here ...
1
vote
1answer
139 views

Maximal “distance” of two probability measures?

Consider the two coins (i.e., probability measures on the discrete set ${0,1}$) $C_{0.9}$ and $C_{0.99}$, where $C_{x}$ is the coin having probability of turning head equal to $x$. Let $\mu_{0.9}$ ...
1
vote
2answers
290 views

Expected Value of Distribution function

So now when I study probability in more mathematical manner I encounter a lack of calculus knowledge especially in Measure theory. So my problem is, knowing that: $I = \int\limits_{\mathbb{R}} ...
1
vote
1answer
6k views

How to find probability density functions?

$X$ is a random variable uniformly distributed on the real interval [0,1]. Through some experimentation, I found that the probability density function, PDF of: $X$ is $1$ or $\dfrac{d}{dx}X$ $2X$ ...
0
votes
1answer
201 views

Is it possible to derive the CDF of $Z$?

Assume that $X_i$, $Y_k$, $i=0,\ldots,N$, $k=1,\ldots,K$ are non-negative independent non-identically distributed random variables. Let us define the random variable $Z$ as \begin{align} ...
0
votes
0answers
69 views

stationary process is invariant under sliding time

$(\Omega,\Im,P) $ is a probability space and $\xi(t)\equiv\xi(w,t)$ is a stochastic process which is defined on $\Omega\times T$ ,$T=[0,\infty)$. If for every ...
0
votes
1answer
292 views

Expected value for the number of goals in a game

I'm trying to use odds data from bookmakers to estimate the expected number of goals in a game. We have these known facts: P(o4.5) = 0.573 P(o5.5) = 0.458 P(o6.5) = 0.279 P(o4.5) is the ...
0
votes
1answer
310 views

Discrete Random Variable Transformation

i have a problem and i can't figure out any solution. Suppose i have this game: i throw a die untill i get a 6. Every time i throw the dice i pay -1 and when i get the 6 i win 5. (Nb: when i obtain ...
0
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1answer
359 views

conditional probability of normal random variables

I have the following set of equations: $$ y = \alpha x + \epsilon $$ $$ z=\beta y + \mu $$ and $\epsilon$ is N(0,$\sigma_{1}^2$), $\mu$ is N(0,$\sigma_{2}^2$) and $x$ is N(0,1) and they are all ...
4
votes
3answers
881 views

Is it a characteristic function?

Can anyone explain, how can I prove either $\phi(x) = |\cos t|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.
0
votes
1answer
185 views

CDF integration question

I am solving a problem where $X$ is an exponential random variable and $\lambda=\frac{1}{10}$. I need to find the CDF of $X$ and have that $\int_0^\infty \frac{e^\frac{-x}{10}}{10}$ turns out to be ...
0
votes
2answers
238 views

How can I calculate the CDF of this random variable?

$X_1$, $X_2$, $X_3$ are random variables distributed following non-identically independent exponential distribution. The PDF $X_i$, $f_{X_i}(x)$=$\frac{1}{\Omega_i}\exp(\frac{x}{\Omega_i}), ...
2
votes
1answer
624 views

what is the intuition behind Delta method?

I'm trying to learn the delta method in probability but couldn't quite get the hang of it. For example: trying to solve a problem from the book statistical Inference : Consider a random sample from ...
1
vote
1answer
379 views

Average Value of Bounded Normal Distribution

Suppose a truck has a capacity of 100 and order sizes to be filled are normal distributed with mean 95 and standard deviation of 10. There is about 30% chance that capacity is exceeded. In this case ...
7
votes
1answer
1k views

Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
3
votes
0answers
158 views

Bayesian inference on partitioned multivariate Gaussian

My question is on Bayesian inference of partitioned multivariate Gaussian. To make things easier, suppose there is a 2-dimensional Guassian, $$ X_1 \sim N(\mu_1, \sigma^2_1) \\ X_2 \sim N(\mu_2, ...
1
vote
0answers
64 views

Sampled or discretised Gaussian Random Variables

Suppose I have $X$, a normal random variable with mean $\mu$ and variance $\sigma^2$. Now I discretise this random variable to form a discrete random variable $Y=g(X)$. $Y$ could be created by ...
1
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0answers
153 views

Expected ratio of successes of Bernoulli provided total number of trials is N

My question seems similar but not exactly explained by negative binomial If I stop experiment when either k successes is reached or after N experiment What is the expected ratio success/required ...
0
votes
1answer
300 views

Density of truncated normal distribution?

I have a truncated normal distribution with mode $0$ and variance $\sigma^2$ that only consists of non negative values. What is the density of this distribution at some non negative $x$? I have just ...
0
votes
1answer
708 views

How to prove $\frac{(n-1)S^2}{\sigma^2}\sim \chi_{n-1}^2$? [duplicate]

Possible Duplicate: Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$ If $X_1,...,X_n$ is a random sample from a normal distribution and ...
1
vote
1answer
956 views

finding variance from a piecewise function

How do you calculate the variance of a piecewise function? For example, what would be the variance of the probability density function $f_x(x)= \frac{3}{4}, 0\leq x\leq 1; \frac{1}{4}, 2\leq x \leq 3; ...
-1
votes
1answer
128 views

expected value problem

Say we have a probability density function $f_y(y)=3y^2$, where $0\leq y\leq1$ and we take 15 observations at random. If $x$ is a number within the interval $(.5, 1)$ what is $E(x)$?
3
votes
1answer
341 views

Convergence in Probability $\Rightarrow$ Convergence in Expected Value

Under which conditions the Convergence in Probability implies the Convergence in Expected Value?
1
vote
2answers
157 views

hypergeometric proof

I've been looking at hypergeometric probability problems and came across this equation. Can someone explain to me why ...
2
votes
1answer
132 views

Given $Z_1$ and $Z_2$ independent normal variables, find a pair $X_1,X_2$ with correlation $p$

I am given $Z_1$ and $Z_2$ independent standard normal variables, I have to find two random variables $X_1$ and $X_2$ with correlation $\operatorname{corr}(X_1,X_2) = p$, where $p\in (−1, 1)$. Any ...
2
votes
1answer
227 views

Parameter estimation for a distribution by minimizing its conditional entropy

Let $X$ be a discrete random variable with Laplacian distribution with mean $0$ and scale $\lambda$, as $$ p(X) = \frac{1}{2\lambda} \exp\left(-\frac{|x|}{2\lambda}\right), \\ X \in ...
0
votes
2answers
172 views

Find $P(X\gt Y)$ using the joint density

$f_{X,Y}(x,y) = \frac{2}{3} (x+2y)$ for $0 < x < 1, 0 < y < 1$; find $P(X\gt Y)$. I got 1/9 by evaluating $$\int_0^1\int_0^{x-1} \frac{2}{3}(x+2y) dy dx$$
0
votes
1answer
602 views

Binomial Distribution with mean and variance

let x1 x2 x3 x4 be random sample from the population that satisfies an binomial distribution with n = 3 and p = 1/4 a) Find the mean and variance for Sum Y = x1 + x2 + x3 + x4. b) Find the mean and ...
0
votes
1answer
151 views

can chernoff bounds be used for proving upper bounds as well as lower bounds

I have a hw problem where it is asked to show theta(n) using chernoff bounds. I am able to prove for O(n) but not in the reverse way.Is it possible to prove both bounds using chernoff?
1
vote
2answers
140 views

Exponential variables

Suppose we have two exponential random variables $X_1$ and $X_2$ with parameters $\lambda_1$ and $\lambda_2$. Would the sum of them have any recognized distribution? If they have the same parameter ...
1
vote
1answer
86 views

A simple inequality in probability

I need to prove this seemingly simple inequality. If $X$ and $Y$ are iid discrete random variables, how does one prove that $$2P(|X-Y|=0)\ge P(|X-Y|=x)$$ where $x$ is any other positive integer. Is ...
3
votes
3answers
5k views

Finding a CDF given a PDF

The PDF for $Y$ is $$f_Y(y) = \begin{cases} 0 & |y|> 1 \\ 1-|y| & |y|\leq 1 \end{cases}$$ How do I find the corresponding CDF $F_Y(y)$? I integrated the above piecewise ...
3
votes
1answer
146 views

Estimating number drawn from one distribution based on sum of that number and number drawn from another distribution

I have been working on this for several days and have been unable to come up with an answer. The problem is very simple to state, but it seems difficult to solve. A computer draws a number $x$ at ...
3
votes
1answer
1k views

How does a function acting on a random variable change the probability density function of that random variable?

Given a random variable $X$ with probability density function $P(X)$, and given a transformation function $f(x)$, how does one determine the new resultant probability density function: $P(f(X))$? For ...
0
votes
1answer
83 views

Vector of normal distributes random variables

If I have $n$ random variables $X^n=(X_t^{(n)})_{t\ge 0}$, all $X^i$ normal distributed and they are independent. Now I define new processes: $$Z_t:=X^{(1)}_t+\dots+X^{(n)}_t$$ Since $X^{(i)}$ are ...
1
vote
1answer
136 views
2
votes
3answers
118 views

A probability distribution and random variable

Assume we have $X_1,...,X_n$ independent poisson random variables. What is the cdf or pdf of $ \sum_{i=1}^{n} X_i$ ??
2
votes
2answers
750 views

Computation of the probability density function for $(X,Y) = \sqrt{2 R} ( \cos(\theta), \sin(\theta))$

Let $R$ be a almost surely non-negative continuous random variable with absolutely continuous measure, and $\Theta$ be an independent random variable, uniformly distributed on the interval $[0, 2 ...
0
votes
1answer
278 views

Finding probability distribution of a random variable from the others

Let $X_1,...,X_n$ be independent exponential distribution.and now i want to calculate the density function of $\sum_{i=1}^{n} X_i$,i tried to find its distributing function,$F(T(X)\le x) $ but then i ...
1
vote
1answer
392 views

PDF/CDF and expected value of a function

How can I compute the PDF/CDF and expected value of the following function: $$ \frac{\alpha}{r^2} $$ where $r$ is generated as follows: draw $x$ and $y$ from a uniform distribution in the range ...
2
votes
1answer
325 views

Understanding the Kesten Multiplicative Process

I read in D. Sornette's Critical Phenomena in Natural Sciences about the Kesten Multiplicative process: $$X_{n+1} = a_n X_n + b_n$$ Where $a_n$ and $b_n$ are stochastic variables drawn from the pdfs ...
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vote
1answer
183 views

The Nature of Probability Mass/Density Functions

Consider a certain random variable and all its possible probability mass functions (or probability density functions). What structure does this space have? For example, it can be endowed with a ...
3
votes
1answer
1k views

coin flips and markov chain

Consider the case of an infinite (or finite $n$) string of coin tosses, and let $q$ and $1-q$ be the probabilities that the coin comes up tails and heads, respectively. (For simplicity, we can take ...
2
votes
0answers
111 views

expectation of rademacher chaos

Let $\sum_{i=1}^n\sum_{j=n+1}^m\epsilon_i\epsilon_jb_ib_j$ be Rademacher Chaos of degree two (here $b_k\in R$ and $\epsilon_k$ are Rademacher random variables) and such tat ...
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1answer
194 views

Multinomial distribution: probability that at least one variable takes a certain value

Let $(X_1,\ldots,X_M)\sim \operatorname{Mult}(N;p_1,\ldots,p_M)$ follow a multinomial distribution. What is the probability that at least one of the variables takes a certain value, i.e. ...
0
votes
1answer
229 views

A problem of bivariate normal distribution

suppose $(X_1,X_2)\sim\mathcal{N}(0,0,1,1,\rho)$. find distribution $$\mathbb{Y}=(X_1^2-2\rho X_1X_2+X_2^2)$$
0
votes
1answer
61 views

the expectation of a Normal r.v conditioned by possible observation

I got stuck with this problem, Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$, where $\sigma^2$ is known and $\mu$ is unknown and will be updated using Bayesian inference. We give ...
2
votes
0answers
95 views

The expectation after Bayesian inference of a Normal r.v

I'm confusing myself with this question. Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$. We known the variance $\sigma^2$ however don't know the mean $\mu$, and choose to use ...
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0answers
319 views

Generating an asymetric triangular distribution

I am trying to generate an asymetric triangular distribution; a is lower limit, b is higher limit and c is mode. I found this following way to generate a random variable $X$ with triangular ...
1
vote
2answers
3k views

Determining p-th Quantile From Probability Density Function

I'm not sure how to derive the $p$-th quantile. I know that it is point which divides the distribution of $X$ into two parts, but I'm not sure what I'm supposed to do here. If the random variable ...