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

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0answers
92 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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2answers
133 views

Probability of the highest order statistic below the population median. [closed]

What is the probability that the highest order statistic of a random sample of size n from any continuous distribution is below the median ( population median ) of that distribution.
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1answer
28 views

Probability: How much days we need to play a game win

Suppose the probability of win a lotery game is : $1/1000$ If a person play the lotery every day with the same combination, how much time he need to wait to win the lotery? Im thinking to use a ...
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1answer
31 views

Multivariate sampling of $F(x_1,…,x_n)$?

Let $$(X_1,...,X_n)\sim F(x_1,...,x_n)$$ (not independent). How can I sample from this distribution? In the univariate case, on can use $F^{-1}(u),u\sim U(0,1)$. However, in the multivariate case ...
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2answers
411 views

Prove that if $X$ is stochastically larger than $Y$ then $E(X)\ge E(Y)$

Prove that if $X$ is stochastically larger than $Y$ (i.e. $P(X > t) \ge P(Y > t)$ then $E(X)\ge E(Y)$. I understand how to solve the problem if $X$ and $Y$ are non-negative random ...
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0answers
125 views

A property of the hazard function of the normal distribution

I have a problem that I can't figure out. Define $$\Gamma\left(x\right):=\frac{\phi(x)}{1-\Phi(x)}$$ where $\phi(x)$, $\Phi(x)$ are the density respectively cumulative distribution function of the ...
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0answers
91 views

question about exponential distribution or exponential random variables

Consider a post office that is run by two clerks. Suppose that when Mr. Anderson enters the system he discovers that Mr. Smith is being served by clerk 1 and Mr. Brown by clerk 2. Suppose also that Mr....
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2answers
173 views

Show that Y=aX+b is an random variable. [closed]

Let X be an random variable on a given probability space and let a,b∈R. Show that Y=aX+b is an random variable. if X has a distribution function F, what is the distribution function of Y? if X ...
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0answers
36 views

What is the limiting distribution of this Markov Chain?

Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$: Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the ...
2
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0answers
26 views

4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as $μ=E({\bf z})$, $Σ=E({\bf z}{\bf z}^H)$...
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3answers
439 views

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement.

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement. Let $X$ be the number of yellow balls on the first two draws and $Y$ the number of ...
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0answers
39 views

Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = \...
2
votes
1answer
75 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(X=0)$.Here is my solution: $$\mu= \frac {p(2) 2!}{p(1)}$$ Then how can find the mean? Thanks.
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1answer
46 views

Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$

Let $U$ have a uniform distribution on $[0,1]$. Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$ My attempt: $F_Y(x)=P[Y\le x]=P[{1\over ...
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0answers
39 views

What is the transformation that maps a Gaussian distribution to a Beta distribution?

Suppose X is a random variable with Gaussian distribution over domain $\mathbb{R} = (-\infty, +\infty)$, with PDF function $f_X$. And Y is a random variable with Beta distribution over domain $[0,1]$,...
2
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1answer
83 views

Probability of Sample Variance Given Variance

I am trying to solve a problem that I have never seen before and cant seem to find a way to solve it so any help or tips would be appreciated! Here's the Problem: Suppose a considerable amount of ...
2
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0answers
479 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P\left(F^{-1}(F(X)...
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1answer
51 views

Shortcut to finding $E(XY)$

The question says "Find $E(Y|X)$ and hence evaluate $E(Y)$ and $E(XY)$" The joint pdf is $$f_{X,Y}(x,y)=\begin{cases} 8xy, & \text{ for } 0< y< x < 1, \\0, & \text{ elsewhere } \end{...
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1answer
16 views

Distributing dimes to 2 groups of people such that each member of one group gets at least one

I have a study question that I have the answer for, but I just can't understand how or why it is the answer. The question is: $n$ dimes are distributed to $y$ young people and $o$ old people. Every ...
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1answer
125 views

Inequality for $N(0,1)$ CDF: $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$

Suppose that $F$ is the CDF of a standard normal distribution. Hayashi (2000) claims that the following is true $$ |\log F(v)|\leq |\log F(0)|+|v|+|v|^2\quad\text{for all}\quad v. $$ How does one ...
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1answer
140 views

A die is rolled 24 independent times. Let Y be the sum of the 24 resulting values. What is the variance of Y?

A die is rolled 24 independent times. Let Y be the sum of the 24 resulting values. What is the variance of Y? I know how to find the mean of Y, but I'm having some trouble finding the variance of X ...
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1answer
45 views

Is there an interpretation of the Beta Distribution?

There are cases in probability where one distribution has an "interpretation" in terms of another distribution: X ~ Gamma(k,1/m) for positive integer k, can be interpreted as the distribution of ...
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2answers
279 views

Convolution CDF formula?

In reference to this post, the pdf of dependent random variables $A+B$ is given by: $$f_{A+B}(z) = \int_{-\infty}^{\infty} f_{A,B}(a,z-a) \mathrm da = \int_{-\infty}^{\infty} f_{A,B}(z-b,b) \mathrm ...
2
votes
1answer
56 views

Showing that $X_n$ ~ $N(0, a_n)$ converge to $0$ when $a_n \to 0$ sufficiently fast

If $X_n$ have distribution $N(0, a_n)$ with $\sum_{n=1}^\infty a_n^b < \infty$ for some $b > 0$, then $X_n$ converge almost surely to $0$. I was able to show (for a previous part of the same ...
2
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1answer
127 views

If the difference of two i.i.d. random variables is normal, must the variables themselves be normal?

I previously asked a similar question about the sum of two i.i.d. random variables, thinking the two cases to be equivalent. But I can't see how to apply the proof of that case to this one. It is ...
1
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1answer
43 views

Maximum of RVS independent and identically distributed

I am having a small doubt regarding maximum of random variables. I have $$Z= \max\{ X_1, X_2,\dots X_p, \dots X_N\}$$ where all $X_i$ are independent, identically distributed. Now, If for sure, I know ...
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1answer
60 views

Marginal Probability of Stochastic Process

I have a wide sense stationary stochastic process x(t)=asin(2πf0t)+bcos(2πf0t) where a & b are independent gaussian random variables. How can I find the Marginal probability of x(t)? I am ...
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1answer
30 views

what will be the PDF of the magnitude of this random variable x+j y?

if we have a complex random variable [x+j*y] where (j :sqrt(-1)) and x,y both have Gaussian distribution and statistically dependent , so what will be the distribution (PDF) of the magnitude of this ...
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4answers
108 views

Proving two random variables differ with positive probability

EDIT: Despite the help of the posters below, I'm still confused. I'm rephrasing the question slightly. Can someone hep me with rephrased problem: Suppose that $X$ is a random vector and $Y$ a random ...
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1answer
22 views

Prove existence/non-existence of a pdf given mean, std, range

Given: Mean = 100, Range = [4, 10000], std = 3000 Is it possible to prove whether a pdf exists or not that satisfies these values? If it does exist, what would be approximate shape of the ...
2
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1answer
141 views

If the sum of two i.i.d. random variables is normal, must the variables themselves be normal?

It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose $X$ and $Y$ are two i.i.d. random ...
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1answer
106 views

E(XY) = E(X).E(Y|X) . Is this true for mean = zero.

I know that Joint Probability density function for two random functions $X$ and $Y$ $$P(XY) = P(X)\cdot P(Y|X)\tag{1}$$ But I just read in a set of lecture notes that for E(X)=E(Y)=0 $$E(XY) = E(X)...
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1answer
24 views

Distribution of difference of points in same tail of normal curve?

If $x$ and $y$ are random values drawn from the part of a normal curve that is greater than a fixed $C$. The distributions of $x$ and $y$ are clearly not normal, but is the distribution of their ...
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0answers
175 views

Exponential distribution.Bank waiting time

I am at a total loss with that problem: Consider a bank with three tellers and a single waiting line.Every customer of the bank is serviced,when he reaches the top of the waiting queue,by one and only ...
1
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1answer
93 views

Finding MLE, MOM of a distribution

I'm stuck on a particular problem and I'm not quite sure what to do. The problem reads as such: Let $X_1, X_2, . . . , X_n$ be a random sample from a distribution with density $f(x) = \frac{xe^\...
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1answer
45 views

Tail probability of a max of iid

If $X_{i}$ are iid random variables with $X_{i}>0$ and $\mathbb{P}(X_{i}>t)\sim t^{-\alpha}$ as $t\to \infty$. Then my question is: Is it also true that $\mathbb{P}(\max_{1,\dots n} X_{i}>...
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1answer
93 views

Independence of sum and quotient of exponential variables

Let $X_1$ and $X_2$ be independent random variables each following an exponential distribution with the same parameter $\lambda$. (They both have density function $f(x) = \lambda e^{-\lambda x}$ for $...
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0answers
93 views

what will be the PDF of magnitude and phase of this random variable?

i have a random variable as shown in the figure and i tried to find the PDF of the magnitude and phase of this random variable using central limit theory as i mentioned , i know that if we have ...
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2answers
39 views

Tip Top Landscaping Company-word problems

Max has only 9 months and has to take care of 1/4 of the yard, Alex has work for the company longer and has to takes care of 1/3 of the yard, Steven has been with the company the longest and must ...
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1answer
200 views

A die whose score cannot be as before (Markov chains)

A die is "fixed" so that each time it is rolled the score cannot be the same as the preceding score, all other scores having probability $1/5$. Given that the first score is 6, what is the probability ...
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0answers
64 views

Drawing balls from a finite urn (non IID Galton-Watson)

I have an urn containing $N$ balls. I begin a process of drawing balls from the urn by taking $1$ ball from the urn (with probability 1). Balls are drawn without replacement. Each ball that is drawn ...
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1answer
138 views

What is the probability of this question?

On a single draw from a deck of playing cards the probability of selecting heart is 1/4 the probability of selecting a black card is 1/2. what is the probability of selecting both a heart and a black ...
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2answers
62 views

Formula the conditional probability of mables

I have a interesting question that need your help. I have two sets A and B. Set A have 10 marbles that numbered from 1 to 10. Set B have 6 marbles that numbered from 1 to 6. Randomly choose $g$ ...
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2answers
588 views

Find the marginal PMF of X given the following joint PMF

Oh boy, so this is a tough one. Let X and Y be 2 random variables with the joint pmf: $$p(x,y) = \frac{e^{-\lambda}\lambda^{x}p^{y}(1-p)^{x-y}}{y!(x-y)!}$$ $$y=0,1,\ldots,x $$ $$x = 0,1,2,\ldots$...
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2answers
75 views

Relating a Gamma Distribution to an Exponential one?

Question related to Gamma and Exponential random variables. Suppose I have a Gamma random variables with shape and scale parameter $m$ and $\theta$ i.e $$X\sim\Gamma(m,\theta)$$ respectively. Can I ...
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1answer
58 views

Help with Linear Transformation of a multivariate normal

Given X ~ $N_2$ (μ, Σ)$ Find the Distribution of $$ \begin{pmatrix} X+Y \\ X-Y \end{pmatrix} $$ Show independence if $Var(X) = Var(Y)$ Attempt: Given proper of ...
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1answer
203 views

Find a symmetric random walk on $\mathbb{Z}$ that is transient.

I wanted to know if it is possible find a symmetric random walk on $Z$ that is not recurrent. Let $X$ have the following distribution, with a probability $1/2^{i+1}$, $X=\pm b_i$. Let $$S_n=\sum_{k=...
3
votes
1answer
54 views

Where do I go wrong?

Suppose $X,Y$ are independent Uniform$(0,1)$ random variables. Find the probability $P(Y\geq X\mid Y\geq\dfrac{1}{2})$. Please note that I know the correct answer and that I have arrived at the ...
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1answer
23 views

Probability, Minmize Gaussian distribution

There is a one problem that bugs me a while: Two random variables with distribution $X$ is $\mathrm{Gaussian}(\mu=\frac{-3}{\sqrt5}, \mathrm{var}=\frac 9 5)$, and $Y$ is $\mathrm{Gaussian}(\mu=\frac{-...
2
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0answers
117 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$? Let $\varphi_X(t)$ be ...