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

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Mean Preserving Spread and concavity of the discrete function

Could someone help me with the understanding of the following thing? Consider a discrete distribution with pmf $p_k$ and its mean preserving spread (MPS) $p_k'$. Also let the set $a_1, \ldots, a_n$ be ...
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1answer
34 views

Why is the area under the pdf for the Von Mises distribution not one?

I've been playing with the Von Mises distribution for a project I'm doing in python and I'm confused about it. I'm drawing the pdf, which is defined by wikipedia here as $p(x|\mu, k) = \frac{\exp{(k ...
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1answer
39 views

Probability confusing question

I saw this in my probability class past exam papers I saw the answer key but I still can't fully understand. I wish somebody can walk through this with me :) A company takes out an insurance policy ...
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1answer
22 views

Calculating the distribution of a compound random variable

Given $X\sim U(1, 0)$ and $Y\sim Exp(1)$, determine the density function of $Z:=\frac{X}{Y}$. Now, without looking up how to do it I tried to figure it out myself. The value of the density function ...
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25 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if variables are gaussian?

Is the Gaussian random variable with density given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{x^2}{2 \sigma^2}}$$ the unique RV such that the joint pdf of $N > 1$ independent and ...
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8 views

“Distance” of iid gaussian variables

Take two i.i.d. Gaussian R.V.s $X$ are $Y$ both of which are $~N(0,a\sigma)$. Define a new R.V. $D = \sqrt{X^2 + Y^2}$. What's the expected value $E(D)$? In researching this I'm seeing references ...
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1answer
14 views

Joint distribution function from marginals

Is it possible to obtain joint distribution function when only the marginal distribution functions of random variables are given and, the random variables are not independent? If possible, it would ...
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1answer
20 views

Conditional probability distribution notation versus conditional probabilities of a single sample space?

When writing conditional marginal probabilities, the following seems to be the notation: $$p_{i|Y=y_{j}} = P(X=x_{i}|Y=y_{j}) = \frac{P(X=x_{i},Y=y_{j})}{P(Y=y_{j})}=\frac{p_{ij}}{p_{+j}}$$ This is ...
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12 views

Unbiased estimator for geometric distribution parameter p

I believe that the MLE of parameter $p$ in the geometric distribution, $\hat p = 1/(\bar x +1)$, is an unbiased estimator for $p$ and would like to prove it. So far, I have: $E[\bar x + 1] = E[\bar ...
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67 views

Random walk, expectation & variance, joint probability, approximation question [on hold]

Consider the following random walk on a plane: The walk commences at the origin and at each timestep, a step of unit length is taken in a random direction $\theta$ (measured relative to the positive x ...
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1answer
31 views

Expected value with negative exponent

I am trying to solve identify the expected value of a statistic that involves a fraction. I have simplified the expression to: $E[\frac{1}{1+ \sum_i x_i}] = E[\frac{1}{1+ T}]$ However, I am not sure ...
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1answer
37 views

Approximating a joint pdf using normal density of two independent variables

I know that given these two random variables (which correspond to the $x$ and $y$ coordinates of a random walk after $n$ steps, their joint probability density function can be $approximated$ by a ...
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1answer
15 views

Binomial-Poisson limit

I want to show that if $Z_n$ has the binomial distribution with parameters $n$ and $\lambda/n$ with $\lambda$ fixed, then $Z_n $ converges in distribution to the Poisson distribution, parameter ...
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1answer
13 views

Probability function and distribution - taking out fish from a pool

In a pool of fish there are 4 fish of type A, 3 fish of type B, 2 fish of type C, 1 fish of type D. We take out fish without returning them until we get fish of type C for the first time. ...
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1answer
19 views

Binomial distribution tail inequality

Let $X \sim \mathrm{Bin}(n,p)$ does there exist $l$ ideally $l=f(n)$ such that $P(X<l)=o(1)$ in the limit $n\rightarrow \infty$? I'd be looking for the largest possible $l$.
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1answer
15 views

A Continuous random variable X has probability density function $f(x)=ae^{-ax}$

A Continuous random variable X has probability density function $f(x)=ae^{-ax}$ where I found $a=0.5ln2$ I Found that the mean of this distribution occurs at X=2. Now, I was then asked what is: ...
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1answer
33 views

Poisson Distribution to Calculate plane crashes

The number of passenger planes that crash every day follows the Poisson distribution with parameter p. The number of crashes each day is independent. What is the probability of exactly 3 planes ...
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1answer
20 views

How to get the value of 'scaled' binomial distribution?

People kindly told me that there is not a equivalent popular distribution for $aX$ when $X$ is distributed as Binomial, but it is just a 'scaled' distribution. Here, $a$ is a positive constant. ...
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1answer
18 views

Poisson Probability (Shopkeeper Sales)

SOLUTIONS: (A) 0.1804 (B) 0.0166 (C) 0.3233 Mean = 2/7*5 (a) x = 3 (b) x > 5 I'm still unsure how to approach each question, because I still get the wrong answers.
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1answer
31 views

pdf: What is the distribution of aX when X ~ Binomial / Gaussian

Question When $X$ is distributed as binomial or Gaussian, is $aX$ equivalent to some famous distribution? Here, $a$ is a real and positive number. Background I know a general formula giving $aX$'s ...
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1answer
29 views

Joint Probability with many values

Consider I have the following tree structure which provides the relation between various entities. Associated with this, I have the following table with data. ...
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13 views

Comparing results of calculated probability and practical probability [on hold]

I am planning to compare probability that comes from theory and practical experiment. So here the detail of my experiment: I have black box B where there are N lines as input and N lines as output, I ...
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1answer
36 views

Is there any simple formula for this probability distribution of random walk?

Assume $\{S_n\}_{n\geq 0}$ transits as follows: $S_0=0$, for $k\geq 1$, $P(S_{n+1}=k+1|S_n=k)=\alpha$, $P(S_{n+1}=k|S_n=k)=\beta$ and $P(S_{n+1}=k-1|S_n=k)=1-\alpha-\beta$, where ...
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1answer
18 views

Prove that the expected value of a pareto random variable $X$ is equal to$ (\frac {a}{a-1})\cdot \lambda$ [on hold]

I am stuck on where to begin with this problem. Any pointers would be much appreciated.
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1answer
30 views

Expected value of Bernoulli with probability of success Gaussian distributed

I have a circle with centre $(0,0)$. I am generating Matlab code to include $N$ neurons in a neural network. The probability of including individual neurons in a network decays exponentially with ...
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1answer
34 views

What distribution it is based on the histogram? [on hold]

I generated this histogram in r and was trying to determine which distribution I should use, my guess is normal or Binormial. But I'm not sure, can anyone help please?
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17 views

How to show the series of expectations for truncated symmetric random variables is convergent

Suppose that $(X_n)$ is i.i.d. with symmetric distribution and that $E(|X_1|)<\infty$. I want to show that $\sum\limits_{i=1}^{\infty} \frac1iE(X_i 1_{[|X_i|<i]}) $ converges. Attempt: Since ...
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1answer
15 views

Rescaling a probability

I can't ge me head around this. I know that between 00:00h and 00:30h (i.e. within 30 minutes) a person is with a chance of 90% in room A, 7% in room B and 3% in room C. Now the task is, to derive a ...
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1answer
13 views

Finding marginal distribution integration help

Let: $f_Y(y)=e^{-y}$ Let: $ \mathbf P(X=k$ | $Y=y)$ = $\binom{2}{k}(e^{-y})^{k}(1-e^{-y})^{2-k}$ where k = 0, 1, 2 To find the density of $X$: $f_X(k) = \int_0^ \infty ...
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1answer
27 views

Expected value for sum of iid normal variables squared

Let $X_i$ be iid from a $N(\alpha, \alpha)$ distribution. I am trying to find $E[\sum_1^n X_i ^2]$ and thought that I would be able to transform the statistic $\sum_1^n X_i ^2$ into a chi-squared ...
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10 views

Lists of common sufficient statistics

Can someone suggest a source for common sufficient statistics for exponential families? For example, I'm looking for something more comprehensive than the Wikipedia page for sufficient statistics, ...
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1answer
18 views

Poission Distribution [on hold]

An Airport reports that aircrafts arrive at a certain runway according to a Poisson process with a rate λ = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson ...
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1answer
27 views

Random variable and distribution - number of tests a teacher has to make

$100$ students do a test. The probability of failing the test is $0.6$, those that failed, do a retest, the probability of failing the retest is $0.5$. Those that fail the retest do another retest. ...
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0answers
16 views

pdf for the sum of squared iid normal random variables

I am trying to find the distribution/pdf for the sum of squared $X_i$ iid observations from the normal distribution $X_1 ,..., X_n$ ~ $N(\alpha , \alpha)$, i.e. $X_1 ^2 + X_2 ^2 +...+ X_n ^2$. I was ...
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10 views

What is the distribution of 'max of some normaldistributions'?

Suppose I have two random variables $a$ and $b$. $a$ follows a normal distribution of parameters $u_1, s_1$. $b$ follows a normal distribution of parameters $u_2, s_2$. $u_1$ and $u_2$ are the ...
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Can I use Geometric Distribution to find the law of a total?

I have a variable X which is the amount of minerals in a dL(deciliter) of water. X follows a Normal Distribution X~N(μ,σ). I have the probabilty of the P(a ≤X< b) in a dl, where a and b are ...
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1answer
22 views

Probability Distribution sampling problem

$\text{*The below problem was asked in geometric distribution section}$ In a population there are $50\%$ Male and $50\%$ Female What is the probability to find $2$ Females in a row out of $10$ ...
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12 views

Normal distribution and covariance matrix

The question is as follows: Consider a normal distribution with mean $\mu = 0$ and covariance matrix $\sum$. Let $v$ be an eigen vector of $\sum$ with eigen value $\lambda$. What does it mean ...
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47 views
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Find probability distribution knowing the conditional probability distribution

I have been working on some physics problem which I "translated" to the following mathematical problem for which I need help to solve: Suppose we have the random variable $\alpha$ distributed ...
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1answer
11 views

Maximizing Varience of Independent Random Variables [on hold]

Suppose X and Y are independent mean 0 random variables, with positive variances a and b, respectively. Find the value of c that minimizes the variance of cX+(1-c)Y?
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1answer
46 views

Finding the variance of the time series defined as $x_t=\phi x_{t-1}+w_t$, for $t=2,3,4,…$.

Let $w_t$ be white noise with variance $\sigma_w^2$ and let $|\phi|<1$ be a constant. Consider the process $x_t=w_1$ and $x_t=\phi x_{t-1}+w_t$ for $t=2,3,...$. I need to find the variance. I ...
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19 views

Stationary distribution of an increasing stochastic process with a cut-off

I have a discrete time stochastic process $\{X_t : t \in T\}$ with continuous state space. Assume $X_0=0$ and increments $\delta_t$ are exponential with mean $\alpha$ (so its parameter is ...
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27 views

Random variable: $X\sim Normal(m, {\sigma}^2)$, find the characteristic function of $X^2$

Is it possible, knowing that $X$ is a random variable with normal distribution( with parameters $(m, {\sigma}^2)$), to find the characteristic function of $X^2$ ? What I thought is: Since: $\phi(X) ...
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1answer
24 views

Determine the probability density function of…

Let $X$ be a random variable with normal distribution with parameters: $$m = 1$$ and $$\sigma = 2$$ How can the probability density function of $$Z = -\frac{\ln |X|}{3}$$ be determined?
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20 views

moment generating function with Taylor series simplification

Reading a research paper on the appendix of theorem 2, we try to find moment generating function. Denotes $a(0,r_1,r_2)$ as the annulus with radii $r_1<r_2$ centered at the origin $0$ Consider ...
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1answer
21 views

find distribution of $\max(x^2,x)$ and $\min (x,1)$

I have the following question. Find distribution of $Y=\max(X^2,X)$ and $Z=\min(X,1)$. My distribution function is $$ F_X(x)=\left\{\begin{array}{ll} 0 & \mathrm{if}\; x <0\\ 0.5x & ...
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19 views

Geometric Distribution

The police have stated that 20% of the items sold by pawn shops in the city have been stolen. Ralph has just purchased 4 items from one of the city’s pawn shops. Assuming the official is correct, and ...
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1answer
13 views

Find marginal probability distribution of $ X$?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X= 5\ mL,\ \sigma_Y= 2\ mL, \ \mu_X= 120\ mL, \ \mu_Y= 100\ mL$ and $\rho = 0.6.$ How do I find the marginal probability distribution of ...
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1answer
31 views

Chebyshev's inequality for 1 standard deviation results in 0?

In applying Chebyshev's inequality to a probability distribution, the following is the given equation: $$p(\mu - c*\sigma \le X \le \mu + c*\sigma) \ge 1 - \frac{1}{c^2}$$ This indicates for any ...