This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...

learn more… | top users | synonyms (2)

0
votes
0answers
3 views

Obtaining the density of a Compound Poisson Process using Fourier Inversion Formula

If $X_t=\sum_{i=1}^{N_t}J_i$ and $E(e^{itX_t})=e^{\lambda t (E(e^{itJ_1})-1)}$ Using the Fourier Inversion Formula, $f(x)=(1/2 \pi))\int_{-\infty}^{\infty}e^{-itx}e^{\lambda t ...
1
vote
2answers
14 views

Prove that markov chain is recurrent

I have the following markov chain : $S=\{0,1,2,3\}$ $p_{i,0} = q$ (if we are in one of the states $0,1,2,3$ we can return to $0$ with probability $q$) $p_{i,i+1} = 1-q , i\in\{0,1,2\}$ (if we are ...
0
votes
0answers
19 views

Expectations of squared sum question

I can't seem to figure out why these expectations turn out the way they do, I am currently studying about the Fisher Information. If $X_1,X_2,...,X_n $ are all iid Poission($\lambda$) , then going ...
1
vote
0answers
17 views

Bayes theorem example

For the first question I get a very small probability 3.55%. I use bayes theorem to calculate it, is it correct? It seems a bit small to me. Medical testing Let us imagine that it is discovered that ...
-1
votes
2answers
36 views

Conditional Probability Question.

A letter is known to have come from either 'TATANAGAR' or 'CALCUTTA'. On the envelop just two letters 'TA' are visible. What is the probability that the letter has come from (i) TATANAGAR (ii) ...
0
votes
1answer
20 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 $10$ ...
1
vote
0answers
22 views

Average difference between two odd numbers of equal length

If I select two different odd numbers of binary length $l$, what is the formula that will tell me the average difference between those two numbers? Note that the high order digit must always be $1$, ...
0
votes
0answers
29 views

$2014$ has about a $90\%$ chance of being warmer than $1998$?

According to NASA, $2014$ has a $38\%$ chance of being the hottest year, $2010$ has a $23\%$ chance of being the hottest year, $2005$ has a $17\%$ chance of being the hottest year, and $1998$ has ...
0
votes
0answers
7 views

Kernel estimate in boundary point

Good moorning, I wonder how to prove that if $X_{1}, \ldots, X_{n}$ are iid from exponential distribution with expected value 1, then the expected value of its kernel density estimator in zero is ...
-1
votes
0answers
9 views

Network theory_probability [on hold]

Please help me to understand the probablity i^' sdirect contact N_i (g)={j≠i │ij∈g},of size n_i (g).The size of g is n(g)=∑_(i∈N)▒(n_i (g))/2. Players loose their ob with beakdown probability ...
2
votes
1answer
32 views

Sum of two normal numbers need not be a normal one

Using the translation invariance of Lebesgue measure how to show that sum and difference of two normal numbers need not be normal ? Normal number in $(0,1]$ is a number $\omega$ such that $\lim_{n ...
0
votes
0answers
12 views

Ant walking on a coordinate plane

I'm going to raise the difficulty of the original question one dimension, so maybe a refresher will be good... Link: http://puzzling.stackexchange.com/questions/10839/ant-walking-on-a-number-line I ...
0
votes
1answer
18 views

Throwing a fair die 5 times

You throw a fair die 5 times. What is the probability that the minimum of thrown numbers is 3? I would have said that all possibilities are $6^5$ and that I have $(1*4^4)*5$ ways to get a minimum of ...
0
votes
1answer
10 views

Finding binomial probability, bernoulli trials

The following table lists World Series Lengths for the fifty years from $1926$ to $1975$. Test at the $0.10$ level whether these data are compatible with the model that each World Series game is an ...
0
votes
1answer
17 views

$\Pr(X+Y\geq1)$

Two random variables X and Y have the following joint pdf: $$f_{X,Y}(x,y)\begin{cases}10x^{2}y & 0<x<1,0<y<x\\0 & \text{otherwise}\end{cases}$$ I am asked to find the marginal pdf ...
2
votes
2answers
33 views

What are the odds that two people are friends in a network of 20 people?

If person $A$ has 10 friends and person $B$ has 5 friends, and they are in a network of 20 people, what are the odds that persons $A$ and $B$ are friends? I first thought to divide into cases ...
0
votes
0answers
22 views

Is it necessary to normalize likelihood within an event space before further multiplication among events?

Say I have observed data, and parameters $A,B$: Parameter $A$ contains possible values: $a_1,a_2,a_3$ Parameter $B$ contains possible values: $b_1,b_2,b_3$ Now, assume I know the likelihood of ...
-2
votes
0answers
18 views

In tennis, the probability of a player winning a point on serve given serve statistics. [on hold]

How can I calculate the probability, $p$, of a player winning a point when serving given: The percentage of first serves that the player gets in. (I'm not sure this is relevant/needed). The ...
4
votes
1answer
40 views

Roll eleven dice such that the product is prime

So the problem is: What is the probability of rolling eleven dice such that their product is prime. The dice is numbered from 1 to 6 and there is an equal chance of getting each number. So in order ...
3
votes
1answer
27 views

Probability and cards

A box contains 900 cards enumerated from 100 to 999 (Each number appears once and just in one card). I took some random cards without looking at them and calculated the additions of the digits in each ...
0
votes
2answers
15 views

Probability Multivariate Distributions

A computer generates two independent fixed numbers from a uniform distribution on the range [0,100].Calculate the probability that the first fixed number exceeds the second by at least 20. I'm ...
1
vote
1answer
54 views

Probability of triangle to be acute?

Suppose that someone randomly picks $3$ points $A, B$ and $C$ on a fixed circle. What is the probability of triangle $ABC$ to be acute?
0
votes
1answer
14 views

Finding the y-coordinate of the peak in a gaussian distribution?

First off all, my general understanding of gaussians is not very good, and I'm having issues getting my head around this because I cannot find an explanation of them I can understand. I'm working ...
-3
votes
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?
1
vote
2answers
34 views

Showing that the Lindeberg CLT Condition Holds

Suppose we have a sequence of random variables, $\{X_{n}\}_{n\geq 1}$ satisfying: $\mathbb{P}(X_{j} = 2^{j}) = \mathbb{P}(X_{j} = -2^{j}) = \frac{1}{2}$ Then is it true that the CLT holds? Or ...
1
vote
1answer
18 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 ...
0
votes
2answers
37 views

Help me find $P(A \cup B')$ under the given conditions

I was assigned the task to solve this problem by mathematics teacher which I can't solve because it doesn't make sense to me (I think that it is impossible to solve it). There was an error please try ...
-1
votes
0answers
13 views

Random variables set representation in the sample space

Consider that I have two Random variables $ X : \Omega \rightarrow \mathbb{R} \space , Y : \Omega \rightarrow \mathbb{R}^d$ belonging to the same sample space and a measurable function $\space f : ...
-1
votes
0answers
25 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, ...
0
votes
0answers
26 views
1
vote
0answers
25 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) ...
3
votes
1answer
51 views

Probability - Poisson arrival of rain

I'm trying to solve this Poisson problem. A rain shower lasts 10 minutes and in that time deposits $10^6$ raindrops over 100 $m^2$. a) What is the probability of at least one drop landing in 1 $cm^2$ ...
1
vote
1answer
22 views

Serial Number in a Geometric Distribution

I won't bother posting the whole exercise.Basically, we've got 2000 pc's and 12 of them are malfunctioning. At some point, the exercise writes: We choose the pc's until we find a malfunctioning ...
1
vote
1answer
21 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?
1
vote
1answer
10 views

Conditional Probability of Poisson Variables

I have two independent Poisson variables $X$ and $Y$ with parameters $\lambda$ and $\mu$, respectively. I defined $Z=X+Y$ and found that $Z$ is also Poisson-distributed with parameter $\lambda + \mu$. ...
2
votes
2answers
33 views

Probability mean,variance and standard deviation formula confusion.

I have a confusion in the formula attached. Why and how are the two formulas equivalent ? sigma in the image is the standard deviation of a distribution...
1
vote
1answer
25 views

About the equivalence of two asymptotic probabilistic statements

Let $g(n)$ be some monotone increasing function of naturals, and let $X_n$ be a sequence of positive random variables. Consider the following two claims: Claim 1. $\exists f=o(g(n)),\ ...
0
votes
0answers
16 views

First order moment of multivariate Gaussian random vector

Let $X = (X_1,\dotsc, X_n)$ be a random vector distributed as a multivariate Gaussian with mean $0$ and covariance $\Sigma$. What is $\mathbb{E}[X_1\dots X_n]$?
2
votes
3answers
67 views

Can the probability of an event be an irrational number?

I am wondering whether it is possible to construct an experiment, where the probability of occurrence of an event comes out to be an irrational number.
0
votes
0answers
18 views

Find distribution of rv X_N where N is independent rv and each X_i~exp(\lambda_i)

First time attempting to use MathJax... Excuse my messy question. Question reads: Let $X_1,X_2,\ldots,X_n$ be independent random variables such that $X_i\sim\exp(\lambda_i)$ such that if $i\neq j$ ...
0
votes
1answer
41 views

Proof of infinite monkey theorem.

I was just wondering, does the infinte monkey theorem also has a proof? And why is this called a theorem? It is sheer common sense. And what are its applications. I have heard about PHP and IEP and I ...
0
votes
1answer
17 views

Width of Gaussian distribution from N trials of coin tossing

What is the width of the Gaussian distribution that is generated from performing $N$ trials of coin tossing? Example: In a trial of 1000 tosses of a coin, $P(H)=0.5$ and $P'(H)=0.5$, where $H$ refers ...
0
votes
0answers
13 views

Sample complexity of coin bias problem

I am reading a paper involving learning in Multi-armed bandit case (its okay if you don't know what that is. Just trying to give context here.) To give sample complexity lower bound, they reduce their ...
2
votes
2answers
48 views

Find the probability of solutions of an equation.

Let $x+y+z=20$. What is the probability that all the solutions are distinct? (No two variables have the same value). Assuming that the solutions are only positive integers or zero. I have tried- ...
0
votes
1answer
15 views

$X \sim N(0, \sigma_1^2)$, $Y \sim N(0, \sigma_2^2)$, $U = X+Y$. What are $E[X|U], E[Y|U]$?

$X \sim N(0, \sigma_1^2)$, $Y \sim N(0, \sigma_2^2)$. X, Y are independent. $U = X+Y$. What are the values of $E[X|U], E[Y|U]$? I understand $E[X|U] + E[Y|U] = U$, but I'm not sure how to move ...
-5
votes
0answers
48 views

REALLY tricky Probability question [on hold]

Here is a board game. $$ \longleftarrow \text{left} \qquad\qquad\qquad \text{right} \longrightarrow$$ $$\bigg| \text{win} \bigg| -2 \bigg| -1 \bigg| \text{start} \bigg|\ 1\ \bigg|\ 2\ \bigg| ...
0
votes
2answers
22 views

Mutual information expressed as Kullback-Leibler divergence

My lecturer defines the mutual information: $$ I(X;Y\mid Z) = D_{KL}\big(p(X,Y\mid Z)\parallel p(X\mid Z)\;p(Y\mid Z)\big)$$ Is this correct? I feel like it doesn't really make sense to say that; ...
1
vote
2answers
38 views

Adding probability of multiple dice rolls

Can anyone tell me what are the odds that stage 4 will be reached?: Stage 1: roll a 20 sided die results must be 13 or lower Stage 2: roll a 20 sided die results must be 13 or lower Stage 3: roll ...
-2
votes
0answers
8 views

Multi-step probability problem. Noise and Stochastic Processes. [on hold]

Please see the image below! I am having issues with this problem and would love a solution.
0
votes
0answers
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 ...