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

learn more… | top users | synonyms

2
votes
1answer
17 views

Joint probability distribution

$Y_1$ and $Y_2$ are jointly distributed with density $f(y_1,y_2)=4y_2^2 \qquad 0 \leq y_1 \leq y_2 \leq 1$ Determine the following: $P$( max {$Y_1,Y_2$} $< 1/2$) $P(Y_1+Y_2 < 1/2)$ ...
0
votes
0answers
22 views

Let $X_n>0$ be iid and $P(X_n>t)\sim t^{-\alpha}$, show that $Y_n=n^{-1/\alpha}S_n$ and $1/Y_n$ are tight.

We are given that $X_n>0$ be iid with common distribtuon $X$, and $P(X>t)\sim t^{-\alpha}$, I need to show that the scale of $Y_n$ is $n^{1/\alpha}$. Or in other words show that ...
0
votes
2answers
6 views

Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of ...
0
votes
0answers
9 views

Kth moment of the standard deviation from a normal 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 Kth moment of T about the origin, and state the condition for the existence ...
-1
votes
2answers
12 views

Probability of the highest order statistic below the population median.

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.
-6
votes
0answers
39 views

why do we believe what we taught in 6 grade. [on hold]

...and will to defend an indefendable point. Like the obvious logical falacy of lack of, say, lack of vitamin C causes scurvy...where's the logic...maybe if the limes were given chococlate chip ...
-2
votes
1answer
19 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 ...
0
votes
1answer
13 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 ...
0
votes
1answer
36 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 ...
1
vote
0answers
22 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 ...
1
vote
0answers
12 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 ...
0
votes
2answers
42 views

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

Let X be an random variable on a given probability space and lrt 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 ...
0
votes
0answers
15 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 ...
1
vote
0answers
8 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 ...
0
votes
3answers
23 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 ...
1
vote
0answers
13 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) = ...
1
vote
0answers
21 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
0
votes
1answer
17 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 ...
1
vote
0answers
6 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 ...
1
vote
1answer
20 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 ...
1
vote
0answers
23 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(F^{-1}(F(X)) ...
0
votes
1answer
35 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 } ...
1
vote
1answer
13 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 ...
2
votes
0answers
13 views

Prove $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$ for $F$ is the standard normal CDF

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 ...
2
votes
0answers
15 views

Distribution of the square of magnitude of a Nakagami random variable [on hold]

Given the random variable $$h \sim \operatorname {Nakagami} (m,1)$$ $$ f_{h}(h)= \frac{2m^m}{\Gamma(m)} h^{2m-1} \text{exp}(-m h^2)$$ What is the distribution of the following function $$g:=|h|^2$$
0
votes
1answer
44 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 ...
1
vote
1answer
18 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 ...
0
votes
0answers
33 views

Exponential.Damaged lamps replaced and not replaced

The problem: In a system there are 10 lamps installed.Their life cycle of every lamp is exponentially distributed with mean 5000 hours. A)If all the damaged lamps are being replaced immediately with ...
0
votes
0answers
15 views

Expected Value of Continuous Random Walk

I'm currently attempting my MMath master project. However, i'm a little stuck on an expected value of the continuous distribution. Its where i wish to change my random walk from a continuous walk to ...
0
votes
1answer
14 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 ...
1
vote
0answers
22 views

Questions regarding a Gamma distributed Random Variable ( first moment and square density)

Consider the following Gamma distributed RV $$\operatorname{Gamma }(m_S,\theta_S)$$ with the following shape and scale parameters $$m_S = \frac{(\theta_1+\theta_2)^2}{\theta_1^2+\theta_2^2}$$ ...
2
votes
1answer
30 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 ...
0
votes
0answers
21 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
vote
1answer
21 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 ...
0
votes
1answer
24 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 ...
0
votes
1answer
11 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 ...
2
votes
3answers
60 views
+100

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 ...
1
vote
1answer
15 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
votes
1answer
33 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 ...
0
votes
1answer
53 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) = ...
0
votes
1answer
17 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 ...
0
votes
0answers
5 views

Asymptotic confidence interval

Let x1, x2, ..., xn be a random sample with a density function given by $ f(x) = \frac{3}{\theta^3} x^2 I_{(0,\theta]}(x)$ where $I_{(0,\theta)}(x)$ is the indicator function and $\theta > 0$ ...
0
votes
0answers
12 views

Estimator for a pointfunction depending on a random variable

PROBLEM STATEMENT: Let $X$ be random variable in $m$ dimensional space. The distance between each pair of vectors $x_i^m,x_j^m$ is $D_{i,j}^m =d(x_i^m,x_j^m)$. Correlation Sum, $C(r)$ represents the ...
0
votes
0answers
55 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
vote
1answer
30 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) = ...
-2
votes
2answers
19 views

Having trouble finding Probability distribution function [on hold]

Kindly answer this question. I have a trouble making function. Find the probability distribution of the sum of the dots when two fair die are thrown, also find the graph of distribution function of X. ...
2
votes
1answer
28 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} ...
0
votes
1answer
32 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 ...
0
votes
0answers
17 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 ...
0
votes
2answers
10 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 ...