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

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Conditional Probability with Normal Distributions

Let's say that I have $3$ random normal variables, $A$, $B$ and $C$. They all have a standard deviation of $17.526$, while $A$ has a mean of $143$, $B$ of $139$, and $C$ of $129$. I want to ...
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1answer
41 views

Sum of random variable

Considering two continuous random variables $X$ and $Y$ with $d.f \; F_X, F_Y$ I want to fin the distribution and distribution function of the sum $Z=X+Y$. \begin{align} P\{Z \leq z\} &= P\{X+Y ...
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2answers
20 views

Calculating Probabilities for a cumulative distribution function within a given inequality

Given that K = 1/36, I require some help understanding (b) • Pr(1/2 ≤ X ≤ 1) Is re-written as such: Pr(X ≤ 1) - Pr(X < 1/2) I do not understand why! Is it because Pr(X ≤ 1) is solved as ...
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1answer
12 views

Calculating Probabilities using a cumulative distribution function

For (b) Pr(X greater than or equal to 2) = ? The textbook says as such but I am confused: Pr(X greater than or equal to 2) = 1 - pr(X less than 2) I do not understand why they re-write the ...
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0answers
10 views

Product of Gaussian random variable with hermition of another independent gaussian random variable. [on hold]

If X∼ CN(0,1) and Y ∼ CN(0,1). X and Y are vectors independent of one another. How to find the E[(X†)Y]. What will be the probability density funtion of Z, If Z = (X†).Y ?
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1answer
26 views

To get the skewness and kurtosis directly from probability density function or histogram

This is my first question here. Please understand even if my question is not very clear. I have tried to calculate skewness and kurtosis directly from probability density function (PDF) without ...
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2answers
25 views

I'm not sure if I'm supposed to use a Poisson distribution or Conditional Probability (or both) to answer this question

I have a question that I'm trying to solve. I have the answer but I don't know how they arrived at the answer so I can't compare my work and see where I went wrong. The number of injury claims per ...
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0answers
23 views

help check the derivation of joint pdf of the sample covariance with Hermitian circulant structure

The question comes from evaluating the covariance matrix and its moments of samples received by a circular array. the covariance matrix is proved to have Hermitian circulant structure. Suppose the ...
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2answers
10 views

Expected Profit for Binomial Variable

Part (a) I am familiar with: (a) P(batch is rejected) = P(X greater than or equal to 3) and n = 15 and p(defective) = 0.1 This gives me the correct answer of 0.1841 I am stuck at part 2! I have ...
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3answers
41 views

Calculating expected value for a Binomial random variable

How do you calculate $E(X^2)$ given the the number of trials and the probability of success? $E(X) = np$, then $E(X^2) = $? Do we have to draw up a table for $n=0,1,2,\ldots,n$ and then use the ...
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2answers
29 views

The relation of $P(X=x+1)$ and $P(X=x)$ in binomial distribution

If I substitute the values to the binomial probability theory, it appears as such $${n \choose x+1} p^{x+1} (1-p)^{n-x-1}$$ I don't know how to move on... What am I doing wrong, or are you ...
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1answer
43 views

Why is this a geometric distribution?

For a random variable $X$, $$P(X = x) = (p-1)/p^{(x + 1)}$$ where $p$ is in $(1,\infty)$. Why is $X$ geometrically distributed? (and why would this make it true that $E[X] = 1 / (p - 1)$ ?) I know a ...
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0answers
19 views

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ [on hold]

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ Can you help me with tips and bibliography... I don't understand very good the topic, and I can't ...
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1answer
30 views

Transformation of random variable

I want to prove the following: $$\text{Let F be a distribution function of any random variable $\\$ and G(x) the quantile function (or inverse) of } \frac 1 {1-F(x)}$$ $$\text{Then, for a standard ...
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0answers
36 views

Why does $p(x) = \int d\theta \ p(\theta, x) = \delta(x-X)$

I am reading a probability book and at some point, the following equation comes up: $$p(x) = \int d\theta \ p(\theta, x) = \delta(x-X) $$ where $\delta$ is the Dirac delta. Why is this true? I ...
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1answer
24 views

Distribution and Probability Distribution

I'm studying on the book of Kolmogorov and Fomin: "Elements of the Theory of Functions and Functional Analysis". I'm into the measure theory and I finished the Theorem of Radon-Nikodim. Now finally I ...
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1answer
43 views

Integral of a bivariate normal cdf

Let $$ \Phi_2(x,y;\rho):=\int_{-\infty}^y\int_{-\infty}^x \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{1}{2(1-\rho^2)}(s^2+t^2-2st\rho)} \, ds \, dt $$ be the joint cdf of bi-variate normal random ...
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0answers
14 views

Maximizing the uniformity of density function subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
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1answer
55 views
+50

Mean of Piecewise function resting on IID random variables

Suppose IID random variables $X_t \sim X$ with support on $[0,1]$ and continuous CDF $F(\cdot)$. I wish to compute the expected value (mean) of the a piecewise function with form $$ \Phi (x,\mu) = ...
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2answers
69 views

A limit of an Integral

Consider the following limit $$K=\lim_{x\rightarrow \infty}\frac{1}{x(1-x)}\left(1-\int_{\mathbb{R}}g(y;x)^x f(y)^{1-x}\mathrm{d}y\right)$$ where $f$ and $g$ are any continuous probability density ...
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1answer
19 views

Expectation of an exponentiated quadratic form

Given a multivariate normal random $n\times 1$ vector $X \sim N(\mu,\Sigma)$, what is the expectation $$\mathbb{E}[exp(X^TAX+b^TX)]$$ where $A$ is a $n\times n$ matrix and $b$ is a n-dimensional ...
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0answers
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probability distributions [on hold]

![ Question 1. 1. Using the probability distribution table, what is the value of P(X = 2 or X = 0)? X 0 1 2 3 4 5 P 0.3 0.05 0.1 0.15 0.15 0.25 P(X = 2 or X = 0) = _____ (Points : 1) ...
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1answer
25 views

How to determine long-run probability using conditional probability?

How to determine long-run probability on a calculator and manually? For example: Ben plays a tennis match every day. If he wins on one particular day, the probability that he wins the next day is ...
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1answer
18 views

Distribution of two-sided boundary stopping time of Brownian motion.

If $B_t$ is a Brownian motion, and a one-sided boundary stopping time is given by: $\tau_a=\inf\{t:B_t=a\}$ the distribution of $\tau_a$ is given by: $f_{\tau_a}(t)=\frac{|a|}{\sqrt{2\pi ...
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0answers
23 views

Calculate the CDF and PDF of Y [on hold]

X has an exponential distribution with parameter λ and Y = 2X. X ~ N(0; 1) and Y = X2 X ~U(0,1) and Y=-log(X). (Here, "log" is the natural logarithm so Y= -log(X) is equivalent to X=e^-Y)
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1answer
41 views

Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is $\mathcal{N}(0,t^3/3)$. ...
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2answers
36 views

Random Variable Problem with unrestricted Parameters Worded Problem

I have no idea how to go about solving (a) -> (c) For (a) Is $k=0.2$, because $\frac{k}{1-0.8}=1$ Hence, $P(Z=z) = 0.2(0.8)^x$ But How do we determine the mean or variance with unrestricted z ...
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1answer
16 views

Expectation of Random Variable - Probability Worded Problem

The part I am confused with is (c) I found part (a) which is: p(0) = 7/24, p(1) = 21/24, p(2) = 7/40 and p(3) = 1/120 How do we find the values for a and b, for part (c) ?
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1answer
30 views

Expanding the expected value

How to expand: $E(Y+1)^2$ my working out: $E(Y^2)+E(1^2) = E(Y^2)+1$ (I'm not sure why this is though..) Can someone link to or list the rules for expanding the expected value ......
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2answers
19 views

Finding values of a constant in a probability distribution

A probability distribution for the random variable $X$ is defined by: $$\mathbb{P}[X=x] = K\cdot(0.9)^x,\quad x = 0,1,2,\ldots$$ It is asked to find $\mathbb{P}[X\geq 2]$. When there is a domain for ...
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1answer
17 views

Open-ended Bernoulli distribution

I've found myself puzzled by the following simple discrete distribution: open-ended Bernoulli distribution, which I will now define. The distribution has 2 parameters: $p$, the success probability, ...
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0answers
19 views

Ross probability models questions [closed]

I am studying for a course and have no professors to talk to live, so I hope some members here can be kind enough to help me. Rather than writing everything out, and splitting it up into different ...
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2answers
80 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
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2answers
42 views

Limit of a probability distribution function times $x$

Let $p(x)$ be a probability density function (i.e. non-negative, integrating to 1). Assume further that $\displaystyle\lim_{x\to\pm\infty}p(x)=0$. Is it always true that $$ ...
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2answers
49 views

Parity of the sum of consecutive Bernoulli random variables

$\newcommand{\Var}{\operatorname{Var}}$I have $X_1,X_2,\ldots,X_{n+1}$ i.i.d. rv, each $X_i$ is a Bernoulli rv with parameter $p$, i.e. $X_i \in \{0,1\}$, $P(X_i=0)=1-p$ and $P(X_i=1)=p$ with $0 \leq ...
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0answers
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Distribution of a quadratic form

Let $A$ be a symmetric positive definite matrix, and $x$ a random vector. Suppose we know the distribution of $x^\top A x$. What can we say about the distribution of $x^\top x$?
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0answers
14 views

An example of $k$-independent distributions.

I'm trying to better understand the idea of $k$-independence in distributions. The idea is that a distribution $\mu$ over $\{0,1\}^n$ is $k$-independent if any restriction of $\mu$ to $k$ variables ...
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1answer
29 views

Conditional probability with a normal distribution

Given that Y and L are normally distributed, the expectation of L given Y is $\mu (Y)$ and the variance of L given Y is $\sigma ^2 (Y)$, why is the conditional probability $P(L > x| Y) = \Phi ...
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2answers
28 views

characteristic function of $\sum_i^N X_i$, $N$ is a Poisson distribution

I have a series of $X_i$ random variables, identically and independent distributed. $S_n=\sum_i^N X_i$, with $N$ which has a Poisson distribution and is independent from $X_i$. I have to compute the ...
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2answers
36 views

Distribution of numbers in everyday life

If you were to read tomorrow's newspaper it is intuitively more likely that the whole number 1 would appear more times than 643689443. Is there an expected distribution of numbers used in general? ...
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2answers
30 views

Conditional probability for random variables with different distributions

Random variables $X$ and $Y$ are independent, where $X$ is exponentially distributed with parameter $1$ and $Y$ has uniform distribution on $[-1,1]$ interval. Find $\mathbb{P}(Y>0|X+Y>1)$. My ...
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1answer
15 views

expected value product dependent random variables

My question is strictly operative, if I have, for instance, two random variables $X$ and $Y$, $X$ is a $\mathcal{N}(m,\sigma^2)$ and $Y=e^{h(X-m)-1/2(h^2\sigma^2)}$. $E[Ye^X]$ is $\int y e^x p(x) ...
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0answers
28 views

Normal approximation with dependent variables

I have a sequence of $N$ dependent random variables $$y_i = \frac{x_i}{||\vec x||_2} \quad \mathrm{for} \quad \vec x \sim \mathcal N(0,\mathbb{1}_N),$$ where the $x_i$ are the iid elements of $\vec ...
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1answer
44 views

Poisson, Gamma distribution example.

Can someone explain me answer for these questions? Suppose customers arrive at a store as a Poisson process with λ = 10 customers per hour. The Poisson process of X ∼ Poisson(λ) the time until k ...
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Random walk probability [closed]

A particle executes a simple unrestricted random walk on a straight path, a step to the right of length 1 occurring with probability 1/3 and a step to the left of length 1 occurring with probability ...
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1answer
14 views

normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
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1answer
54 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
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2answers
38 views

Soccer and probability distributions

The USA soccer team is going to play a championship with 7 other tems. The 8 teams, are going to be divided in two groups of 4 each one. From the participants, Brazil is considered the strongest team ...
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0answers
24 views

Testing for the independence of random variables

In probability theory, $X$ and $Y$ are independent if: $f_{X|Y}(x|y)=f_X(x)f_Y(y)$ If I have sample $Y_1,...,Y_n$ and I would like to test if $Y_i$ is independent from the rest of the sample, I ...
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0answers
16 views

Combination of exponential distribution and geometric distribution

I am trying to figure out the distribution times for dark times for the following process. An atom is prepared in state 1 (dark) and decays to state 2 with characteristic time scale T. From state 2 ...