0
votes
1answer
9 views

Probability: Random Variables and Probability Distributions

1) The function: $F(x)=k(1-(1/2)^{[x]})$, $x > 0$ Is the distribution function for a discrete random variable X. Here, [x] denotes the integer part of x (i.e., the greatest integer less than or ...
1
vote
0answers
18 views

$U$-Uniform$(0,2\pi)$, Z-Exp$(1)$, $U$ and $Z$ are independent. Then $\sqrt{2Z}\cos U$ and $\sqrt{2Z}\sin U$ are independent standard normal.

Given that $U$-Uniform$(0,2\pi)$, Z-Exp($1$), $U$ and $Z$ are independent. Show that $\sqrt{2Z}\cos U$ and $\sqrt{2Z}\sin U$ are independent standard normal variables. Thanks in advance for any ...
0
votes
1answer
16 views

Exchangeable/Independent Bernoulli Distribution

Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having ...
0
votes
1answer
28 views

Probability of sample mean [on hold]

A town has $500$ real estate agents. The mean value of the properties sold in a year by these agents is $\$800,000$ and the standard deviation is $\$300,000$. A random sample of $100$ agents is ...
0
votes
0answers
31 views

Probability and Expected profit

I really need help for this qn! You are asked to determine the profitability of a new line of sunglasses, which will retail for \$10. The fixed cost of setting up the line is \$2000. The total number ...
0
votes
2answers
26 views

Flipping several biased coins

Assuming I'm flipping $M$ biased coins with different probability for heads $p_i, i=\{1,...,M\}$. What is the probability of having $k$ times head? Is there a distribution function known for this?
0
votes
2answers
33 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
0
votes
0answers
41 views

Make the sum of random variables converge, while the sum of the variances diverges

Suppose $X_n$, $n=1,2,3,...$, are independent and $Var(X_n)$ is uniformly bounded by finite constant $C>0$. Construct $X_n$ such that $\sum_nX_n$ converges a.s., but $\sum_nVar(X_n)=\infty$.
0
votes
1answer
32 views

Pdf of the product of an exponential r.v. and a beta r.v.

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
1
vote
0answers
30 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
0
votes
2answers
28 views

Ratio of Gamma random variables

If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$. Attempt: Let $Y_1 = \frac{X_1}{X_1+X_2}$ and ...
1
vote
1answer
21 views

Finding distribution of random variable if X is exponential $(1)$

Let X be an exponential (1) random variable, and define Y to be the integer part of X+1, that is $\hspace{15mm}Y=i+1$ if and only if $\hspace{5mm}i \leq X \leq i+1, i = 0,1,2,...$. Find the ...
3
votes
1answer
44 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
1
vote
1answer
42 views

Pdf of the product of an exponential rv and a $f_Y=Ka^{-K}y^{K-1}$ distributed rv …

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ has the following Pdf: $f_Y=Ka^{-K}y^{K-1}, 0 \le y \le a $. Someone claims ...
1
vote
0answers
23 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
0
votes
2answers
51 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
0
votes
2answers
20 views

Conditional probability in multinomial distribution

Consider a multinomial distribution with $r$ different outcomes, where the $i$th outcome having the probability $p_i$, $i$=1,...,$r$, $\sum_{i=1}^r p_i = 1$. Denote $X_i$ be the number of times the ...
2
votes
1answer
33 views

Using the inverse Gaussian integral to find percentiles

I need some help with the following: Let $$R=\mu+\sigma*\epsilon \hspace{1cm} \epsilon \sim N(0,1)$$ I want to argue that $$ \mu + \sigma*\Phi^{-1}(u)$$ are the percentiles of the model when ...
0
votes
3answers
60 views

Is it even possible to find the variance of this moment generating function?

This is my moment generating function: $M_x(t) = \frac{6e^t}{t^2} + \frac{6}{t^2} + \frac{12e^t}{t} - \frac{12e^t}{t^3} + \frac{12}{t^3}$. I have to find the mean the variance of it. After taking ...
1
vote
1answer
36 views

An inverse problem on tail probability

This is a question out of curiosity. Assume that $f(x)$ is a density function for which there is a constant $C>0$ so that $$ \int_t^\infty f(x) dx \le C f(t) $$ holds for large enough $t>0$. My ...
1
vote
1answer
25 views

Is there a shorter way for me to answer this joint probability question?

Suppose a box contains 10 green, 10 red, and 10 black balls. We draw 10 balls from the box by sampling with replacement. Let X be the number of green balls, and Y be the number of black balls in the ...
3
votes
2answers
47 views

Random variable $X^2$ determined by moments

Let $X$ be a real random variable, with standard normal distribution. Is the distribution of $X^2$ determined by its moments? In general, if $n \in \mathbb N$, is the distribution of $X^n$ ...
1
vote
0answers
22 views

Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
0
votes
0answers
18 views

Simulating beta random variables

Let's say I estimated the parameters of Beta distribution with a single-period dataset, and want to generate a multi-period sample. How can I do that? To be more specific, I estimate my parameters ...
0
votes
1answer
22 views

Derivative of integral over part of Gaussian distribution

I am currently trying to compute the following derivative and integral: $$ P\psi_\theta = \frac{d}{d\theta}\int_{-k}^k tf_T(t)dt, $$ where $t=x-\theta$ and $X\sim N(\theta_0,\sigma^2)$. $f_T$ above ...
0
votes
1answer
22 views

Variance from the pdf

FOr the interpretation of the variance, it is the fluctuation of the data around the mean. So if I know that mean (say mean=0), and then there are lots of data (70%) points are greater than +/-10 away ...
-2
votes
0answers
32 views

How to prove that the distribution function Fx is left continuous if and only if the distribution law µ is non atomic [closed]

How to prove that the distribution function $F_x$ is left continuous if and only if the distribution law $\mu$ is non atomic. Can the law $\mu$ and lebesgue measure be singular if the distribution ...
1
vote
1answer
69 views

Explanation of how probability density functions transform under the change of variable

I've just read about probability density function from this article. In that article, there is some wired concept that I can't understand, please see the section named "Dependent variables and change ...
0
votes
1answer
13 views

Related to chi-squared functions

I'm finding difficulty in finding what type of function it is in continuous distributions in probability.Mainly how can i identify whether a function is chi-squared or not?
1
vote
2answers
39 views

Continuous and Discrete random variable distribution function

I have a very basic question in probability. It pertains to the difference between a continuous random variable distribution function and a discrete one. This question has confused me many times. ...
1
vote
1answer
40 views

Probability: basic question and concept

I have always been struggling with the problem, in particular, I usually have great difficulty in differenting when should I multiply n! to take care of the ordering, and when should I not do so. For ...
1
vote
0answers
18 views

Finding $a_n, b_n$ so that a sequence converges in distribution to a nondegenerate random variable.

Now, $X_1, X_2,\dots$ are iid with the same distribution as the chi-squared distribution with one degree of freedom. Find $a_n$ and $b_n$ so that $a_n \left( \max_{1 \leq i \leq n} X_i - b_n \right)$ ...
1
vote
2answers
33 views

What does tightness of convergence mean and why do we use this?

In a paper that I am reading it is written: We have $$(X(t_j))_{j=1}^k \xrightarrow{d} (Y(t_j))_{j=1}^k,$$ where $t_0=0, t_1 < \dotsb < t_k$. It further is written: In order to show that ...
0
votes
0answers
26 views

Native algorithm for lottery

Consider a simple lottery game that you are required to pick 6 numbers out of 50 numbers (1 to 50), and you have the history of the most recent n games' result, by ...
1
vote
2answers
40 views

Correlation of random variables with joint PDF proportional to $x^{a-1}y^{b-1}(1-x-y)^{c-1} $

The random variables $X$ and $Y$ have joint PDF $$f(x,y)= \frac{\Gamma(a+b+c)}{\Gamma(a)\Gamma(b)\Gamma(c)}x^{a-1}y^{b-1}(1-x-y)^{c-1} $$ where $0 \leq x \leq 1 , 0 \leq y \leq 1, x+y < 1 $ where ...
-1
votes
1answer
33 views

finding out the probability density of a random process

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what ...
3
votes
1answer
13 views

$L^p$ integrability of products of Gaussian variables

Gaussian variables have moments of all orders, so by Hölder's inequality the product of two Gaussian variables $\xi$ and $\eta$ has finite $L^1$-norm: $$ \|\xi \cdot \eta\|_1 \leq \|\xi\|_2 \cdot ...
1
vote
1answer
44 views

What is the correct equation for conditional relative entropy and why

I was trying to understand the concept of conditional relative entropy. As in: $$D(P(X\mid Y) ||Q(X\mid Y))= E [\log\frac{P(X\mid Y)}{Q(X\mid Y)}]$$ I would have thought that its equations would ...
0
votes
1answer
14 views

Geometric Random Variable of a coin toss which is tossed 1 time.

The geometric random variable is defined (in this example) as the number of tosses needed for a head (a fair coin) to come up for the first time. $P_x(k)$ = $(1 - p)^{k-1}$ * $p$ So I calculated the ...
1
vote
1answer
23 views

How is this Negative Binomial Random variable used to solve this problem?

I was looking at the solution to this problem below and I don't understand how they used a negative binomial R.V. to solve the problem. A research study is concerned with the side effects of a new ...
0
votes
1answer
25 views

Computing absolute distribution from conditioned probability

for a sum $X:= \sum_{i=1}^n 1_{U_n<U_0}$ of a series of random variables $U_1, ..., U_n$, all of them uniformely distributed on the unit interval as is $U_0$, I computed the following conditional ...
0
votes
2answers
36 views

Probability of transferring a ball from one bucket to other

There are $n$ buckets. Each of these buckets can hold maximum $k$ balls. The probability that a ball is transferred to $m_{th}$ bucket is denoted as $P(m)$. What is the probability that a ball is ...
0
votes
0answers
15 views

True or false: If a distribution has a conjugate prior, then it is a member of the exponential family.

I would like to know if it's true that "A distribution has a conjugate prior if and only if it is a member of the exponential family". I know that all members of the exponential family have conjugate ...
0
votes
1answer
29 views

How do you transform probabilities from the form P[X=x] to P[X =< x]

I'm working on a problem that requires you to use a binomial distribution to solve the problem. Now we want to determine x such that P[X > x] =< 0.01 or, ...
1
vote
0answers
32 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
3
votes
1answer
28 views

Convergence of expected values as random variables converge almost surely

Let I have a sequence of random variables $X_n$ that converges to random variable $X$ almost surely as $n\to\infty$. How can I proof that $\lim_{n\to\infty}\mathcal{E}[X_n]=\mathcal{E}[X]$ where ...
1
vote
1answer
32 views

Can this function be a density function of a continuous random variable X?

F(x) = 0, if x < 1 F(x) = 1, if 1<=x<=2 F(x) = 0, if x>2 I think it could be, as long as the integral is 1. Any ideas?
0
votes
1answer
14 views

How to calculate the median and the quantiles of this distribution?

A median of a distribution of one random variable $X$ of the discrete or continuous type is a value of $x$ such that $P(X < x) \leq 1/2$ and $P(X \leq x) \geq 1/2$. If there is only one such $x$, ...
1
vote
0answers
27 views

Sample Mean and Sample Variance

Consider a sample of data S obtained by flipping a coin x, where 0 denotes the coin turned up heads, and 1 denotes that it turned up tails. S = {1, 1, 0, 1, 0} What is the sample mean for this data ? ...
1
vote
2answers
16 views

Geometric and binomial distribution problem

Let $X \sim Bi(n,p)$, and $Y \sim \mathcal{G}(p)$. (a) Show that $P(X=0)=P(Y>n)$. (b) Find the number of kids a marriage should have so as the probability of having at least one boy is $\geq ...