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 (...

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1answer
24 views

Expected Value of a sum of a sequence

I have encountered this problem and don't know even where to start. Let $ M = \{s_1,s_2,s_3...s_k : s_i \in \{1,...,i\}\} \subseteq \{1,...,k\}^k $ We select a sequence from M in a uniform ...
1
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1answer
51 views

Using the Central Limit Theory to solve $\lim_{n\rightarrow \infty} \Pr(n-\sqrt n \lt X_1+X_2+\cdots+X_n\lt n+\sqrt n)$

$X_1,\ldots,X_n$ are independent random variables that are uniformly distributed between 0 and 2. What is: $$\lim_{n\rightarrow \infty} \Pr(n-\sqrt n \lt X_1+X_2+\cdots+X_n\lt n+\sqrt n)$$ Attempt: ...
0
votes
1answer
27 views

I am trying to find answer to this bivariate normal problem. Can anyone help. [on hold]

The distribution of the heights of husband-wife pairs in a particular population is modelled by a bivariate normal distribution. The mean height of the women is 165cm and the mean height of the men is ...
1
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1answer
32 views

$X_1, X_2, \dots$ uncorrelated, $\frac{Var[X_i]}{i} \rightarrow 0$, then $\frac{S_n}{n} - \frac{\mathbb{E}[S_n]}{n} \rightarrow 0$ in $L^2$

Let $X_1, X_2, \dots$ be uncorrelated random variables with $\mathbb{E}[X_i]= \mu_i$ and $\frac{Var[X_i]}{i} \rightarrow 0$, when $i \rightarrow +\infty$. Show that $\frac{S_n}{n} - \frac{\mathbb{E}[...
1
vote
1answer
20 views

Showing a “signed Markov transition density” will lead to a trivial measure on path space.

Let for all $t>0$, $x\mapsto p(t,x)$ be a Schwartz function on $\mathbb R$, satisfying $\int_{\mathbb R}p(t,x)\mathrm dx=1$ and $\int_\mathbb{R}|p(t,x)|\mathrm dx\equiv C>1$ for all $t>0$ (so ...
-1
votes
0answers
64 views

Independence of sum of independent variables [on hold]

Given three i.i.d. distributed random variables $X$, $Y$, and $Z$, are $X+Z$ and $Y+Z$ still independent? Edit: Is simpler if all RVS are uniformly distributed in [0,1]? I think the PDFs then follow ...
1
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2answers
44 views

There are 10 boxes, 15 balls; 10 red, 5 blue. Each is randomly placed in a box in an independent manner. What's E[X=the number of empty boxes?]

There are 10 boxes, 15 balls; 10 red, 5 blue. Each is randomly placed in a box in an independent manner. The red balls are placed in boxes 1-10, blue balls are placed in 1-6. What is the expected ...
1
vote
1answer
25 views

20 identical balls to be distributed in 3 identical boxes with MAX & MIN balls in each box?

As the title suggests, In how many ways can 20 identical balls be distributed in 3 identical boxes with at most 8 balls in each box and minimum 1 ball in each box ?
3
votes
1answer
47 views

The probability two balls have the same number

Suppose I have $10^6$ jars, and $k$ balls are randomly and independently placed in each jar. I am given that the probability that there exists a jar with 2 balls is approximately $50\%$. Then $k$ is: ...
0
votes
1answer
18 views

Relationship Between EV(time) and EV(Repetition)

Consider an Stochastic Process with Expected value of time of occuring =T (less than infinity) can we deduce that Expected value of number of Occuring until time T is equal to 1?? if not in General ...
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0answers
8 views

Numerical scheme and boundary condition for 2D Fokker Planck equation

I have a 2D stationary Fokker-Planck equation $$\frac{\partial^2 P(A,B)}{\partial A^2}+\frac{\partial^2 P(A,B)}{\partial B^2}=\frac{\partial f_1(A,B) P(A,B)}{\partial A}+\frac{\partial f_2(A,B) P(A,B)...
0
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0answers
17 views

Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)_{...
0
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2answers
24 views

Finding the probability distribution of the sum of geometric distributions? [duplicate]

X and Y are both geometric distributions with success p. What it the Pr(X+Y=n) Would I use a convolution with a sum for this? Do I need to define a third random variable?
0
votes
1answer
54 views

What's the chance to reach 32000 tries on a check if the picked number equals a random number both in the range of 1 to 100

Correction: It's a game we had to code at school last week. The steps: Pick a number between 1 and 100 (1 and 100 included) Take a random number between 1 and 100 (1 and 100 included) Check if ...
2
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0answers
49 views

Probability with various stages

I'm trying to make a probability calculator for a tennis match. I've already managed to get the probability calculation so I can see what are the chances of winning "the game" for the player who is ...
0
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0answers
25 views

Markov Process converges intuition

To explain my question, I think best to start with the example: assume Markov matrix like this: $ 0 < a < 1$ $$ P = \begin{bmatrix} a & (1-a) \\ (1-a) & a\end{bmatrix}$$ The question is ...
0
votes
1answer
25 views

Probability density function for the number of zeroes X in a five digit code?

Five digit codes are selected at random from the set {0,1,2,...,9} with replacement. If the random variable X denotes the number of zeros in randomly chosen codes, then what are the space and the ...
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2answers
35 views

Probability of even sum of $n$ integers with uniform distribution from $\{1,2,\dots, 2n\}$.

Choosing with Uniform distribution $n$ numbers from $\{1,\dots,2n\}$ with returns and the order is important. What is the probability that the sum of these number will be even? Thanks.
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0answers
26 views

would it be possible to measure neural network combination/ representation as capacity?

My question is about machine learning, If you are familiar with you the restricted Bolatzmanne machine. I would like to know if we can implements some ideas. RBM in simpler case is just two layers ...
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1answer
55 views

If a student who received an A in probability is chosen at random, what is the probability that he/she also received an A in calculus?

This question has been asked before but the solution given was incorrect.(see here) A prerequisite for students to take a probability class is to pass calculus. A study of correlation of grades for ...
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0answers
29 views

Intuition behind expression for expected value in terms of CDF [duplicate]

Let $X$ be a random variable with support on $S \subseteq [0, \infty)$. Let $f$ be the pdf of $X$ and $F$ be the cdf. I'm trying to get some intuition behind this identity for random variables with ...
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0answers
30 views

Expected times to sample all the teeth on randomly stopped gear with 292 teeth [on hold]

I have a ring gear with 292 teeth. Each time the gear stops I have access to 97 teeth on the gear through an inspection hatch to inspect them. The gear stops randomly. How many inspections are ...
1
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0answers
32 views

Equivalent definitions for weak/distribution convergence

We let $X$ be a compact metric space and consider $C(X)$ to be the space of all continuous functions on $X$. The dual space of $C(X)$ can be seen as the set of all signed borel measure on $X$. My ...
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0answers
24 views

Calculate MLEs numerically

I have a random variable $r$, which is define as $r=f(C,\beta, U)$, $C$ and $\beta$ are both parameters and U is distributed as uniform[0,1]. I want to get MLEs of C and $\beta$. My $f()$ is quite ...
0
votes
1answer
28 views

Joint Probability between sample and function of sample [on hold]

I'm wondering if the following statement is true: Let X be a random sample of size n from some distribution. Let T(X) be a function from R^n to R. Is the following statement correct: P(X,T(X))=P(X) `
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votes
1answer
53 views

Find the variance of W when given $W = x + 2y + 3z$. [on hold]

x,y,z are random numbers given w = x + 2y + 3z. also given that the mean of x,y,z= 1,8,0 respectively. what is the mean of the random number w ? Assuming the Standard deviation of the random numbers x,...
-1
votes
1answer
42 views

In the answer to the question attached below, I don't quite see how step-3 is derived from step-2, Can anyone explain [duplicate]

Calculating the expected values of the min/max of 2 random variables Consider two fair $k$-sided dice with the numbers 1 through $k$ on their faces, obtaining values $X_1$ and $X_2$. What is $\mathbb{...
0
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0answers
55 views

Problem understanding an example about $\epsilon$-nets

An ε-net (pronounced epsilon-net) is any of several related concepts in mathematics, and in particular in computational geometry, where it relates to the approximation of a general set by a collection ...
0
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0answers
36 views

What is the almost certain convergence of $\sum_{i=1}^n \frac {X_i} n$ if $P(X_n=n^2)=\frac 1 {n^2}$ and $P(X_n=0)=1-\frac 1 {n^2}$? [on hold]

My guess is that it is 0 as $P(X_n=0)->1$ but I can't prove it, the definition of almost certain convergences lead me nowhere.
1
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2answers
44 views

Probability ace on top and ace on bottom of shuffled deck?

Would someone please explain the answer below: What is the probability that a randomly shuffled deck of 52 cards has an ace as the top card and an ace as the bottom card? ANSWER: ( (4 C 1)(3 C 1) ...
0
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0answers
14 views

Why $(d+1)^{2T}{4T+R \choose R}e^{-2R+14T} \le \epsilon$ given $R=cT \ln d + \ln \frac{1}{\epsilon}$?

I read a paper and I found that the only step I don't understand is this derivation from $$R=c T \ln d + \ln \frac{1}{\epsilon} \\\text{to}\qquad (d+1)^{2T}{4T+R \choose R}e^{-2R+14T} \le \epsilon$$ ...
2
votes
1answer
33 views

Can't understand one chance in R of winning where R is some result of factorials.

In lotto game, let you select six no. from 51 no. on a card and the Lotto managers pick six no. at random. If your choice exactly matches theirs, you win a few dollars. If you have to pick 6 values ...
1
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1answer
42 views

Complicated probability question [on hold]

There are 20 empty present boxes numbered from 1 to 20 are placed on a shelf, there are 4 men standing in front of the shelf. Each one asked to pick in his mind 3 numbers without telling any one then ...
0
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0answers
10 views

References request: two-queue, one-server model with pre-emptive queue priority and finite buffers

Sorry of the title is a mouthful. I'm developing a queue model with the following characteristics: Two queues: One contains an infinite number of people (Queue A) while the other (Queue B) is ...
0
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0answers
11 views

Logic-Calculating Cd Failures

I am working on homework and have the problem At a company every 4th CD is tested, the testing consists of 4 testing programs and the probability that they fail are as follow Program 1 : .01 Program ...
0
votes
2answers
46 views

Point A is picked randomly in a circle with a radius of 1, and center O. What is the variance of length OA?

Point A is picked randomly in a circle with a radius of 1, and center O. What is the variance of length OA? I believe the CDF has to found first, then we need differentiate it, find the expected ...
1
vote
1answer
37 views

I am trying to prove the distribution function for the 'birthday problem' can anyone help?

Let $Y_1, Y_2, . . .$ be i.i.d. and uniformly distributed on the set ${1, 2, . . . , n}$. Define $X^{(n)} = \min \{k : Y_k = Y_j \,\,for \,\,some \,\,j < k\} $, the first time that we see a ...
1
vote
2answers
35 views

Choose 3 cards from a deck, if last two are spades what is the chance of first card being a spade?

I'm stuck on this problem, and need some explanation if possible : From 52 cards we take 1, after that we take 2 more, both of which are spades. What are the odds that first card is also a spade? ...
1
vote
1answer
26 views

Consistent Estimator and Convergence Variance

I was practicing an exam, and came along this question: A consistent estimator converges in probability to the true parameter value. Therefore, the variance of such an estimator converges to zero ...
0
votes
1answer
34 views

Probablity of normal distribution when x is a function

Assume a uniform distribution random variable X~U(0,1). And $\Phi$ is the symbol of the standard normal distribution. Assume $Y=\Phi^{-1}(X)$. The question is, $\mathbb{P}(Y \le 0)=?$. The Solution is ...
0
votes
3answers
66 views

Compute $\mathbb{P}(1<X^2+Y^2<2)$ when $(X,Y)$ is i.i.d. standard normal

Assume that $(X,Y)$ is i.i.d. standard normal. Compute $\mathbb{P}(1<X^2+Y^2<2)$. So I've decided to use polar coordinates to solve and I've gotten to this point: $$\iint_{1\lt X^2+Y^2\lt2} ...
5
votes
1answer
42 views

Is it incorrect to call the probability mass function by the name “discrete probability density function”?

Commonly, the probability density function (pdf) is used when dealing with continuous random variables, while the probability mass function (pmf) is used for discrete random variables. This also ...
1
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0answers
91 views

Switching independent experiments: Does $I_A(X_1,\dots,X_n,Y_{n+1},Y_{n+2},\dots)$ converge almost surely?

Suppose there is a sequence of independent random variables $X=(X_1,X_2,\dots)$ with $X_i$ taking values in some arbitrary measure space $E_i$. Now there is a second sequence $Y=(Y_1,Y_2,\dots)$ ...
1
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0answers
65 views

How many possible solutions are there in 4 number game?

$4$ number game consist of $4$ random number from $0$ to $9$. The goal is to make the result equal to $24$. There are only operation four operation possible, addition, subtraction, multiplication, and ...
0
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0answers
40 views

MLE of a distribution

I have a function $r=C(Q-Q^{1-\beta})$, where C and $\beta$ are constants. And I know the density function of Q. The density function of $Q$ is $\theta Q^{-1-\theta}$, where $\theta$ is a constant and ...
1
vote
1answer
35 views

What is the expected number of samples you’ll take until $x_{i+1} > x_i$?

Randomly (but uniformly) sample a number between $0$ and $1$. Stop the first time that a sample is greater than the immediately previous sample. That is, if your samples are labeled $x_1, x_2$, etc., ...
5
votes
3answers
76 views

chances of a group being all of the same sex [on hold]

I was wondering, if there are 10 girls and 10 boys in a classroom, and they were randomly assigned in groups of four, what are the chances of there being a group with all people inside it the same sex ...
1
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0answers
32 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...