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

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1answer
399 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 ...
0
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1answer
48 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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2answers
36 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 ...
1
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1answer
57 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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3answers
63 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
37 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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2answers
36 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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1answer
53 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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2answers
78 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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0answers
50 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
48 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
90 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 ...
2
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2answers
56 views

Independence of two digits in a uniform$(0,1)$ random variable

I having troubles proving this: Let X be a Uniform$([0,1])$ distributed random variable. And let $X_n$ be the nth digit in the decimal expansion of $X$. Prove that if $n \neq m$ then $X_n$ and ...
0
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1answer
44 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 ...
1
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1answer
50 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 ...
1
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1answer
66 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
43 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 ...
0
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1answer
49 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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1answer
37 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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2answers
42 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
27 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, ...
1
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1answer
122 views

Bernoulli Distribution with support different from $\{0,1\}$

Suppose the support of a distribution is $\{12 , 13 \}$ with $P(X = 12) = p$ and $P(X = 13) = 1-p$. Is this still a Bernoulli distribution even if the support is not $\{1, 0 \}$?
4
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1answer
85 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 ...
1
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2answers
51 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
601 views

Finding the distribution of a poisson distribution with random variable lambda

So suppose $X$ is a rv with a Poisson distribution with $\lambda$ being a random variable as well. $\lambda$ has an exponential distribution with mean $1/c$ and $f_\lambda(t) = c\times\exp(-ct)1_{[0, ...
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0answers
19 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 $x^\top x$?
3
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2answers
69 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
21 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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2answers
44 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
53 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 ...
2
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2answers
52 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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1answer
98 views

Transforming distributions

There is an economy, populated by a large number of agents. A first order condition common to all agents, is the following: $$E[\exp^{(1-\theta)\eta_i}(r-R+\eta_i)]=0$$ the index $i$ indicates the ...
0
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1answer
39 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
36 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
151 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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0answers
36 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 ...
0
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1answer
69 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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2answers
54 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 ...
0
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1answer
28 views

Finding the conditional probability from a conditional distribution function

I'm taking a probability theory class and I'm having troubles with multivariate distributions. In particular, I don't really understand how to find conditional probabilities. Here's a question I'm ...
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0answers
39 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 ...
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0answers
29 views

When is Complex Normal Distribution equal to Normal distribution for real numbers

Let $Z = X+ iY$ be a complex random vector with real and imaginary part equal to $X$ and $Y$ respectively. Assuming that $Z$ has complex Normal distribution, can we say that making $Y=0$, the ...
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1answer
61 views

Question about the Bayesian Inference of a parameter

In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf . In order to ...
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1answer
59 views

Probability: Gamma Function vs Gamma Distribution

Could someone help me with setting up the function of this question. I've been setting it up with the gamma distribution function but kept getting the wrong answer. What I did was I used the Gamma ...
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2answers
54 views

Identifying the distribution which represents a negative binomial distribution as a compound poisson distribution

Suppose that the random variable $X$, which has a negative binomial distribution with probability $p$ and parameter $r$, can be represented as the summation of $N$ iid random variables $Y_1, Y_2, ...
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3answers
66 views

Find $P(X+Y\le 0)$ given the joint probability function of $X$ and $Y$

I am struggling with part c of this question. Could someone please tell me how to approach and solve this type of questions?
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2answers
482 views

Proof of, and requirements for, the reverse of Jensen's Inequality for concave functions

As I understand it, Jensen's Inequality states $$\int_{U}f_{V}\left(h(u)g(u)\right)du\geq f_{V}\left(\int_{U}h(u)g(u)du\right)$$ For a convex function $f_{V}$, a probability distribution $g(u)$ on ...
1
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1answer
135 views

Finding the percentile of a normally distributed variable

I'm taking a probability theory class and I'm stuck on a question. Here's the question: A manufacturing plant utilizes 3000 electric light bulbs whose length of life is normal distributed with mean ...
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0answers
67 views

Probability distribution of k consecutive successes with n maximum trials

Let $X$ be a random variable that represents the number of trials of a given experiment. The outcome of a single trial is a Bernoulli random variable, with probability of success $p$, and trials are ...
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2answers
50 views

Probability Distribution, where $E(X^2) = 2E(X)$

May I please get help with this question? What is the answer and how do I get to it? [Within the context of discrete random variables]. Consider a probability distribution where $E(X^2) = 2E(X)$. In ...
0
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1answer
38 views

Total law of probability in continuous space

I am finding little difficulty in the following definition of total probability specified in a NLP related paper. Say $q^i$ is a partition of my continuous sample space. The authors have defined the ...