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

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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
12 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
16 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
12 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
30 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
15 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
29 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
26 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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0answers
12 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
27 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
14 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
19 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 ...
1
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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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0answers
16 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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0answers
12 views

UMVUE for altered Normal distribution

Let $X_1 , ...,X_n$ be a sample from a normal population $N(\mu , \sigma^2)$. It's easy to find the UMVUE for $\mu$ and $\sigma^2$: (1) After finding the joint density of X=$(X_1 ,...,X_n)$, we find ...
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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 ...
2
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1answer
24 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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8 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
26 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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0answers
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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0answers
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7 views

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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0answers
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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0answers
38 views
+100

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
35 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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0answers
9 views

Stationary distribution of a stochastic process

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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0answers
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
23 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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14 views

moment generating function with Taylor series simplification

Denotes $a(0,r_1,r_2)$ as the annulus with radii $r_1<r_2$ centered at the origin $0$ Consider two bands $a(0,s,t)$ and $a(0,u,\sim)$ for $1\leq s\leq t\leq u$ Suppose a variable (call it an ...
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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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0answers
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 ...
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1answer
30 views

Lottery Distributions Question [on hold]

In a certain lottery, $7$ balls are drawn at random (without replacement) from $n$ balls numbered $1$ through $n$. Let $P$ be the probability that no pair of consecutive numbers is drawn. Let $Q$ be ...
2
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1answer
47 views

Random variable with infinite expectation

I was trying to find $Y$,a random variable (non-negative, may be $E(|Y|)=\infty$), such that $$\sum_{n=0}^{\infty} E\Bigl(\frac{|Y|}{n^2 +|Y|}\Bigr)=\infty$$ I tried with Cauchy distribution but could ...
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1answer
13 views

Average Waiting Time for mixed distribution function

Mixed Distribution Function $$ F(t) = \begin{cases} \hfill 0 \hfill & t < 0 \\ \hfill p+(1-p)(1-e^{-yt}) & t \geq 0 \end{cases} $$ How can i find the average waiting time of an ...
2
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1answer
24 views

What is the pdf and cdf of $aX^2+bX$?

If $X$ is normally distributed, $X \sim N(0, \sigma) $, what distribution is $aX^2+bX$? Is there any way to express the cdf and pdf? Thanks.
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1answer
53 views

What is the probability that a multivariate Gaussian random variable is greater than zero?

I am looking for a way to find the probability that $p(x > 0)$, where the vector $x$ has a multivariate Gaussian distribution $$ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \sim ...
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0answers
15 views

Support of Distribution Function

Suppose I have a distribution function $$C(u,v)$$ with domain $I^2$. Let us define the support of this function as the complement of the union of all open subsets of $I^2$ with C-measure zero. Based ...
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10 views

Distribution of $\sum n_i(U_i-U_{(1)})$

Let $U_i$ be independent random variables with pdf $f_i(x)$ ($i=1,\ldots,k$) where $$f_i(x)=\frac{n_i}{\sigma}\exp(-\frac{xn_i}{\sigma}), x>0$$ Let $n=\sum n_i$ and $U_{(1)}=\min U_i$. ...
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1answer
107 views

Discrete distribution problem in medical application

im having trouble with question 2 on one of my math papers. I would greatly appreciate it if someone could help me out here, preferably give me worked out solutions for this question. Thank you for ...
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1answer
26 views

Poisson arrival times joint distribution

The arrival times of the first and second event are $S_1$ and $S_2$, and the number of arrivals follow a poisson process. How would I compute the Joint PDF of $S_1$ and $S_2$? I have found the PDF of ...
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0answers
18 views

Continuous Distribution [on hold]

Suppose we observe the value of a random variable $X$ with pdf $f(x)=2x$, $0<x<1$. The value divides the interval $(0,1)$ into two subintervals. Let $Z= \min[X, 1-X]$ be the length of the ...