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

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Defective Merchandise Probability

If a company currently has a 2% defective rate when making glass bottles, what is the probability that in every 10 glass bottle case, there would be no more than 1 defective bottle? I figured that if ...
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13 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
13 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
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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
12 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
21 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
22 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
15 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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9 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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16 views
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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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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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17 views

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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27 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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27 views

Inequality with poisson r.v. [on hold]

Let $r>0$ and $X \sim Poisson(\lambda)$. Prove that ( $e=2.71...$) $$ \mathbb{E} X^r \le r^r + (e \cdot \lambda)^r $$ I can show it for $r \in \mathbb{N}$ by writing expected value as series, ...
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7 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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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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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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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 ...
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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
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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 ...
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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
45 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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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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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
87 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
25 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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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 ...
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9 views

Maximum likelihood estimates for exponential distribution [on hold]

If the random variable 'x' has the following PDF $f(x)=\beta e^{-\beta (x-\alpha)}, x\geq \alpha, \beta >0.$ What will be the maximum likelihood estimates for both parameters $\alpha$ and ...
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2answers
60 views

Random Variables and Probability Distributions

Little Help here Q-For a laboratory assignment, if the equipment is working, the density function of observed outcome X is f(x) = 2(1-x), 0 < x< 1 otherwise 0 ...
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Flip a coin, then repeat an experiment n times. Show exchangeable but not independent

We flip a fair coin. If it is heads then we roll a die n times, if it is tails we sample a number n times from the set {1, 2, 3, 4} with replacement. We denote the resulting n numbers by X1, ..., Xn. ...
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22 views

The convergence of probability for $X_nY_n$ and $X_n/Y_n$

Suppose that $X_n, Yn$ ($Y_n\neq 0$ a.s) converge to $X,Y$,respectively, in probability. I need to show 1) $X_nY_n \rightarrow XY$ in probability. 2) $X_n/Y_n \rightarrow X/Y$ in probability. My ...
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1answer
18 views

Equivalent of random variable sequences in distribution?

Suppose that $X_n, Y_n$ are sequences of random variable on probability space $\Omega$. If $Xn,Yn$ converges to $X$ ( some random variable ) in distribution, then is $X_n=Y_n$ almost everywhere (a.s)? ...
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20 views

Frequency Distribution and Throughput

I am conducting an experiment on a couple of computer systems but the results I have don't make sense to me. I made each system perform 1000 operations: System A performs operations at a rate of ...
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28 views

To determine asymptotic value of funtion for large N

To show that the function follows normal Gaussian for large value of N (s.t. m is much less than N ) with mean at 'm'. $f(m,N)=\sum_{a=1}^{\lfloor N/2 \rfloor} \binom{N}{S} * ...
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33 views

Distribution of an angle between a random and fixed unit-length $n$-vectors

Suppose I have a random unit-length $n$-element vector $\mathbf{x}$ that is uniformly distributed on an $n$-dimensional sphere, and let vector $\mathbf{a}$ be some other unit-length $n$-element vector ...
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Expectation of inverse of sum of random variables, exponential distribution

I have a question similar to this one: Expectation of inverse of sum of random variables only my variables have exponential distribution. So $X_1, X_2, ...$ are $iid$ with exponential distribution. ...
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1answer
48 views

Expected Payoffs

In simple setting consider revenue is dependent on variable $w$ which is uniformly distributed $[0,1]$. The revenue function is $wd$, where $d$ is development program. How to I get the expected ...
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30 views

The probability that Z is between 0 and -1.61 [on hold]

What is the probability that Z is between 0 and -1.61? Would be good if you could show working. Thanks.
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46 views

There are 20 red marbles, 10 blue marbles, and 5 white marbles in a jar.

There are 20 red marbles, 10 blue marbles, and 5 white marbles in a jar. Select a marble without looking, note the color, and then replace the marble in the jar. We’re interested in the number of ...
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20 views

What's the interpretation of this random variable

Let $(\Omega,\mathscr{F},P)$ be a probability space and $X$ be a random variable that takes values in $\mathbb{N}$. Define $$q(n)\equiv P(X=n)\quad n\in\mathbb{N}$.$$ So $q$ is just the probability ...