0
votes
1answer
22 views

Explicit CDF associated to Gamma PDF [on hold]

Thanks in advance for the help with this! I'm struggling to follow the solution in the book for this problem. Any help is greatly appreciated. Let the distribution function of X for x>0 be: $$F(x) = ...
0
votes
2answers
36 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$(\lambda)$. We let U = X + Y and V = Y, and the ...
0
votes
1answer
29 views

Estimating how much two probability distributions differ

I have two probability distributions A and B. First I would like to estimate how much they differ. In this regard I use as metric the Jensen–Shannon distance (i.e. the square root of Jensen–Shannon ...
1
vote
2answers
36 views

Find $E[N]$, where $N = \min\{n>0: X_n = X_0\}$

Let $X_i$, $i\geq 0$ be independent and identically distributed random variables with probability mass function $$ p(j) = P\{X_i=j\},\; j=1,...,m,\;\sum^{m}_{j=1}P(j)=1 $$ Find $E[N]$, where ...
1
vote
1answer
24 views

Product of two distribution functions.

Let F and G be two distribution functions, does the product FG still a distribution function?
2
votes
1answer
33 views

Expected Value on code

I'm trying to figure out the expected number of times this algorithm will print. I'm stuck on how to go about doing so. I used an indicator variable to keep track of the number of print statements ...
0
votes
1answer
11 views

maximum-likelihood: a sequence of events described by a Bernoulli distribution

I am having quite some troubles with the following homework: In a city it's measured for the whole year whether it rained or not. A distribution $\textrm{Bernoulli}(r_t|\rho)$ characterizes the ...
2
votes
1answer
31 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
1answer
54 views

What is the probability the best case occurs? (Comp Sci Type Question)

I'm having trouble figuring out what's the probability the best case occurs? It's my first time bringing together probabilistic knowledge into computer science. The question goes as such. Consider ...
0
votes
2answers
34 views

Let $X_{1},X_{2}, \dots, \sim Exp(1)$ i.i.d. - Calculate the probability of $P[\max{(X_{1},\dots,X_{n},)} < \log(n) - 5] $ for $ n > e^{5}$

Let $X_{1},X_{2}, \dots, \sim Exp(1)$ i.i.d. - Calculate the probability of $P[\max{(X_{1},\dots,X_{n},)} < \log(n)-5] $ for $ n > e^{5}$ as well as $n \rightarrow \infty $ The correct ...
0
votes
1answer
18 views

For exponential random variables $X_i$, how to find $P(t-X_1<X_2\mid t-X_1<X_3)$?

Assume $X_1, X_2, X_3$ are three independent exponential random variables with means $1/A$, $1/B$ and $1/C$ resp. How do we calculate $P(t-X_1<X_2\mid t-X_1<X_3)$? My try: \begin{align} ...
0
votes
0answers
24 views

Stat problem! Why is this? [duplicate]

This is a statistics problem. although this is not a problem which needs an answer, I want to know the reason Why this is right. Can you guys help me ? Thanks in advance!
1
vote
1answer
30 views

Marginal PMF values of a function

I've been doing this question and I was wondering if my workings are correct, if they are not correct, can you please correct them? The question is as follows: My workings are: $\binom{y}{x} ...
2
votes
0answers
35 views

How can this be derived??

This is a statistics problem. although this is not a problem which needs an answer, I want to know the reason Why this is right. Can you guys help me? A deck of $n=10$ cards is numbered from 1 to ...
0
votes
0answers
11 views

Exercise on Variance Reduction Techniques

I'm struggling with a variance reduction techniques exercise of a book I'm reading. Anyone has an idea of how to approach this problem? Can anyone help me out?
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
1answer
87 views

How do you answer this Bayes theorem question? [on hold]

Your computer is acting strangely and you suspect it has a virus. Unfortunately all 5 of your virus detection programs are outdated. If your computer has a virus then each program, independently of ...
0
votes
0answers
17 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 ...
1
vote
1answer
21 views

Finding the CDF of $g(X)$ where $X$ is a continuous random variable

I imagine this is a rather simple problem, but I'm having a bit of a hard time actually finding the answer. $X \sim \mathrm{Exp}(0.2)$ and $W=g(X)$ given by $g(X) = \begin{cases} X^{\frac{1}{3}} ...
0
votes
0answers
33 views

Joint density of $X_1^2+X_2^2\ \text{and} X_2,\ X_i\sim N(0,1)$

Let $X_1 $ and $X_2$ be iid with a common standard normal distribution. I am looking to find the joint pdf of $Y_1 =X_1^2 +X_2^2$ and $Y_2=X_2$. I know i can use a straight transformation argument but ...
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
27 views

Bivariate distribution question

How would I do this question by hand? I know I integrate from $-\infty$ to $\infty$ for $f_{x,y}$, but I have no idea how to do it by hand! My algebra soup is bad, can someone please help me? P.S I ...
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
1answer
30 views

Question I couldn't identify to solve this distribution [on hold]

A couple decides to have 3 children.If none of them is a girl,they will try again,and if they still don't get a girl,they will try again and continues so on.If X is the number of children,the couple ...
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
36 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
39 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 ...
0
votes
1answer
19 views

Lifetime of Light Bulbs - Probability Question

This is the question that I have, so I solved the first two parts very easily. The first part (i) Then the part (ii) Now, I dont know how to do the final part of the question (it is too ...
0
votes
0answers
6 views

Methods for Uncorrelating data - Comparison

I see that both PCA and Cholesky Decomposition could be used for uncorrelating correlated data. When should one be used? What are the assumptions made by each model. When do the methods fail? Are ...
0
votes
2answers
27 views

What's the PMF of the Difference between 2 Independent Poisson RV?

I searched around and found that the difference between 2 independent Poisson RV $X_1$ (mean $\mu_1$) and $X_2$ (mean $\mu_2$) follows the Skellam distribution such that its PMF is: $$f(k; \mu_1, ...
2
votes
1answer
27 views

Throwing darts at dartboard (cumulative distribution function)

Suppose there is a target shooting game on circle of radius $3$. Think of the result of the shooting as a random experiment, for simplicity, we suppose the hit will always impact on the circle of ...
0
votes
1answer
11 views

$U\sim \mathcal U(0,a)\overset{?}{\implies}U-\lfloor U \rfloor \sim \mathcal U (0,a-\lfloor a \rfloor$)

Suppose $U\sim \mathcal U(0,a)$ for some $a>0$. Is it true that $U-\lfloor U \rfloor \sim \mathcal U (0,a-\lfloor a \rfloor$)? How can I prove this? If $a\in \mathbb N$ then the following ...
0
votes
1answer
28 views

Say (X,Y) has the distribution on the area shown below find P(X>1|Y=1/2) [closed]

Say (X,Y) has the distribution on the area shown below, find P(X>1|Y=1/2)![enter image description here][1]
1
vote
2answers
69 views

uniform distribution over disk

Given two independent random variables $A$ uniform on $[0,1]$ and $B$ uniform on $[0,2\pi]$. Obtain the joint pdf, tranform to the disk, if necessary modify to obtain the uniform pdf over the disk. ...
1
vote
0answers
38 views

Expectation of $\frac{1}{X+1}$ for a geometric random variable

I am confused over $E(\frac{1}{1+X})$ where $X$ is geometric distribution with parameter $p$. The book wants me to prove that $E(\frac{1}{1+X})=log((1-p)^{\frac{p}{p-1}})$ Here's what I did. ...
0
votes
0answers
11 views

Presenting a multinomial dstribution as some function of underlying binomials

I have a multinomial distribution, which arises, let's say, for the sake of clarity, from $N$ rolls of unfair $S$ sided dice and labels on the sides are non-integer. I know the probability for each ...
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 ...
0
votes
1answer
20 views

How to Bayes update discontinuous cdf

I would like to find the probability of an event given a signal (update using Bayes' rule). The events are $x\in \mathbb{R}$ with pdf $g:\mathbb{R}\rightarrow \mathbb{R}_+$, with $g(x)>0\;\forall ...
0
votes
2answers
48 views

Determine the PDF from the MGF

If the moment generating function is given as; $ \psi_X(s) = e^{s^2}$ How can i determine the PDF of $X$?
1
vote
0answers
66 views

Upper bound on the expectation $E[(X-t)^+]$

Let $X$ be a random variable with $E[X]=0$ and $E[X^2]=1$ that satisfies \begin{equation} |F_X(x) - \Phi(x)| \leq \alpha, \qquad \forall x\in\mathbb{R}, \end{equation} where $F_X(\cdot)$ is ...
0
votes
1answer
17 views

Expected Value of a Minimum Function using a Beta Distribution

Let $X$ be a IID random variable with support in $[0,1]$ and CDF given by a Beta distribution, i.e. $X \sim Beta(\alpha,1)$. Suppose we have a function of the form: $$ Z_t = \phi(X_t,y_{t-1}) = ...
-1
votes
1answer
32 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 ...
0
votes
1answer
16 views

What is a time of waiting for 5th success in bernoulli's sequence with p - probability.

What is a time of waiting for 5th success in bernoulli's sequence with p - probability. Hum, what exactly should I found? Should I use Newton distribution for r=5, but what is my k?
0
votes
1answer
23 views

Poisson distribution equation

This is probably a very simple and silly question to ask, but I just don't understand the steps for b). I don't quite understand where the negative (-) sign came from? Could somebody please shed some ...
0
votes
2answers
51 views

Given a CDF, find P(-.5<X<.5)

Given the following CDF: \begin{equation*} F(x)= \left\{ \begin{array}{lr} 0 & x<-1, \\ \frac{x+2}{4} & -1 \leq x < 1 \\ 1 & x \leq 1 \end{array} \right. \end{equation*} Compute ...
0
votes
0answers
28 views

Computing the expectation of following interesting random variable

Every package of some cereal includes a plastic animal. There are $N$ different types of animals, and each package is equally likely to contain any type. Your children make you buy one package of the ...
2
votes
0answers
12 views

Hypergeometric RV - what is the sample/population?

An instructor who taught two sections of engineering statistics last term, the first with 20 students and the second with 30, decided to assign a term project. After all projects had been turned in, ...
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 ...
4
votes
0answers
32 views

Memoryless property and geometric distribution

Suppose $X$ is a random variable taking values in $\mathbb N_0$ with the memoryless property,i.e., for each pair of number $s,t \in \mathbb N$, $$P(X\geq s+t\mid X>t)=P(X\geq s)$$ Show that a ...
0
votes
1answer
13 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 ...