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

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12 views

Distribution of a maximum, where the maximum is a linear combination of the random variable whose distribution is known.

If a variable $q$ distributes $U[0,a]$ where $a = a_0 + h(I)$. In particular $q_i$ distributes $U[0,a_i]$ where $a_i = a_0 + h(I_i)$. What would be the distribution of the following random variable: ...
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1answer
19 views

Finding a probability density function of a function of three dependent random variables

I have three random variables that are functions of another three random variables by pairs, say: $U=fc(X,Y)$, $V=fc(Y,Z)$ and $W=fc(X,Z)$, with $X$, $Y$ and $Z$ being independent random variables ...
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1answer
19 views

Find joint distribution function in region

I can't for the life of me figure this one out, I am stuck on part (c) ... I have this as my starting point ? $$ \frac{45}{304}\int_0^x\int_{2-x}^2 u^2v^2\,\mathrm{du} \mathrm{dv} $$ Here is my ...
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0answers
8 views

Ito integrals and joint distribution with copulas

Let $X_{t}$ and $Y_{t}$ be two brownian motions and let their joint distribution be given by $F$. So in regularly correlated BM's where $dX_{t}dY_{t}=\rho dt$, we have a bivariate normal distribution ...
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1answer
32 views

Normal Distribution Worded Problem

Standard deviation = 2.5 mL 98% of bottles must be between 998 mL and 1000mL Pr( 998 < x < 1000) = 0.98 This is a technology exam question, therefore to find the mean I used the method: ...
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0answers
43 views

Upper bound on the covariance of two gamma processes?

Given two binary gamma processes, $X = \Gamma(t; \gamma_1, \lambda_1)$ and $Y = \Gamma(t; \gamma_2, \lambda_2)$, what is their maximum covariance? Applying this answer, it would seem that it is the ...
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0answers
17 views

Distance histogram within cylinder

Suppose I randomly pick a pair of points $x$ and $y$ from inside a cylinder of radius $R$ and length $L$. What is the probability that they are a distance $d$ apart? In other words, I wish to evaluate ...
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9 views

Perlin noise seems to be distributed like on a bell curve

I coded a perlin noise function, (gradient noise, not value noise). Weirdly, (or maybe it's normal?) my values seem to be distributed pretty much like on a bell curve. I guess I should fix my ...
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3answers
41 views

Probability of $X < Y$ [on hold]

Two independent random variables $X$ and $Y$ have distribution functions $\lambda_1e^{-\lambda_1x}$ and $\lambda_2e^{-\lambda_2x}$ respectively. $Pr(X < Y)$ = ?
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2answers
66 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
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0answers
25 views

Random varible and dicrete probability distribution.

4 unbiased coins are tossed simultaneously. Obtain the probability distribution of the random variable 'numbers of head'.
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3answers
22 views

Median Value + Mode for Hybrid Functions of a Continuous Probability Density Function

To find the median: should I set the integral to 0.5.... but because there are two functions that are non-zero, I am unaware of a method to find the median. To find the mode: would I need to ...
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0answers
37 views

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
47 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
26 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
17 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
11 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
29 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
26 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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2answers
11 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
42 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
30 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
20 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 ...
0
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1answer
31 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
39 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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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
56 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
11 views

probability distributions [closed]

![ 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
26 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 [closed]

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)
3
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1answer
57 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 $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
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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
17 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 ...
1
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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
81 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
11 views

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
30 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 ...