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
29 views

How to find $\lim_{n\to \infty} P(a≤(X_1X_2…X_n)^{-n/2}e^{n/2}≤b)$ where $X_1,X_2,…,X_n \sim U[0,1]$?

I am trying to calculate $$\lim_{n\to \infty} P(a≤(X_1X_2...X_n)^{-n/2}e^{n/2}≤b)$$ in terms of $a,b$, where $$X_1,X_2,...,X_n \sim U[0,1]\,\,\,\,\,\,\,(i.i.d.)$$ and $$0≤a<b$$ My attempt is to ...
0
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0answers
39 views

A student only wrote down …

In the statistics lecture $6$ discrete and $5$ continuous distributions were discussed. For each distribution one can ask for $\mathbb{P}(X = a), \mathbb{P}(X \leq a), \mathbb{p}(X \geq a), ...
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1answer
37 views

Find the Expected value of a Random variable

Assume random variable $$X \sim f_X(x) \,\,\, -2 \leq x\leq 2$$ Now Assume we need to compute the following $$F= \mathbb{E}\left(\frac{1}{1+(G(X))^2}\right)$$ where we define the function $$G(x) = ...
0
votes
1answer
24 views

estimates Gaussian moments

Let $X_i \sim N(0,\sigma_i^2)$. Let $k\geq0$ be a fixed integer. I would like to compute $$A:=E[|X_1-X_2|^k|X_2|^k]$$ My idea was \begin{align*} A=&\int_{\mathbb{R}^2}|x_1-x_2|^k |x_2|^k ...
2
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0answers
14 views

Find the number of items in $10000$ sets of 10 throws each in which you would expect no even numbers.

Given to us is that we have an irregular six-faced die and the expectation that in $10$ throws, $5$ even numbers show up is twice the expectation that $4$ even numbers show up. The question( as in the ...
1
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0answers
16 views

Finding probabilities through normal approximation

Problem: On each day a gambler wins \$1 with probability 0.91 and loses \$10 with probability 0.09, independently of the other days. With what probability will he have a net loss of \$100 after ...
1
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0answers
20 views

Central limit theorem with Lyapunov condition

$Z_1, Z_2,...$ are iid uniformly distributed on $[-1;1]$, $\lim_{n \to \infty} a_n=0$ and $\lim_{n \to \infty} na_n=\infty$ also $a_n>0$ $\forall n$, $X_{n,j}= \frac{1}{a_n}I(|Z_j| \le a_n)$ ...
1
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2answers
24 views

How to make a game with $2$ dice fair?

You pick a number between $2$ and $12$. Then you roll $2$ dice. The result is the sum of the tosses. If your number is not the sum of the tosses then you lose a dollar. If your nmber is the sum of ...
0
votes
2answers
39 views

Correlation between two variables

Assume $X_1$, $X_2$, $X_3$,..., $X_n$ are i.i.d, say that $Y_1$ = $X_1^2/\sum_i X_i^2$ and $Y_2$ = $X_2^2/\sum_i X_i^2$, how to calculate the correlation between $Y_1$ and $Y_2$ or prove that they are ...
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0answers
22 views

Expectation of cumulative probability function

I have a question. Assume there is a normally distributed variable $X\stackrel{d}{=}\mathcal N(0,1)$ and $F(x)$ is cumulative probability function of $X$. There is another variable ...
0
votes
0answers
10 views

probability in investments

an investment analyst collects data on stocks and notes whether or not dividends were paid and wether or not the stocks increased in price over a given period. price increase ...
0
votes
1answer
17 views

Probability issue given a Bayesian Network

If we have a Bayesian Network A -> B ->C then P(B|A, C) = P(B|A)? Thanks!
0
votes
1answer
37 views

Does a conditional normal distribution imply an unconditional normal distribution?

I have often seen it claimed that for scalar random variables $y$ and $x$, the conditional normal distribution $$ y\mid x\sim N(0,x^2) $$ also implies the unconditional normal distribution $$ ...
1
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0answers
8 views

How to write a combination that defines all possible edges in a graph?

Given a graph $G=(V,E)$, I would like to define a set that contains all possible edges in the graph where the edges can't be repeated. In other words, if the graph has three nodes $x,y,z$ then I want ...
1
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2answers
50 views

If $\{X_n\}$ converges in probability to $1$, where does $\{1/X_n\}$ converge to?

Without using the continuous mapping theorem, I want to show that, given $\{X_n\}$ is a sequence of random variables converging in probability to $1$, $\{1/X_n\}$ converges in probability to $1$. The ...
3
votes
3answers
23 views

The probability of selecting both defective items when taking 10 out of 24

The following is a problem from my probability text. A box contains 24 light bulbs, of which two are defective. If a person selects 10 bulbs at random, without replacement, what is the probability ...
-1
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0answers
13 views

Chain conditional probability issue [on hold]

In conditional probability network if A -> B ->C then P(B|A, C) = P(B|A)? If no, then what is the answer? Thanks!
0
votes
1answer
14 views

Is it possible to evaluate a normalizing constant for a characteristic function

Let $X$ be a random variable with density $f$ and characteristic function $\varphi$. Say we know $\varphi$ up to a constant $c$. Is it possible to evaluate this constant using $\int f(x)dx=1$ (or by ...
0
votes
6answers
79 views

How come everyone says that you can't with in lottery because of statistics yet every single day I hear that someone has won?

I'm a very simple man with basic understanding of mathematics and theory. This question has bugged me for the last few years, ever since I learned about lottery tickets. When I talk with people about ...
0
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1answer
23 views

Which is the difference between $P(A \mid B)$ and $P(A=t \mid B)$ in a Bayesian Network?

Which is the difference between $P(A \mid B)$ and $P(A=t \mid B)$ in a Bayesian Network, where $A$ and $B$ are boolean values?
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0answers
21 views

Conditional probabilities given the evidence(Bayesian network)

Let's say we have a Bayesian network: How can I compute P(A | F, E) ? I have all the probabilities for each node. Thanks!
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0answers
14 views

Combinations of inheriting genes with certain variables

Context. The idea is taken from a breeding mechanic of a game similar to inheriting genes. The variables are highlighted in bold and italicized. There are 6 stats from each parent represented by 6 ...
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0answers
24 views

Borel Sets and relation to probability theory.

I am currently having difficulty understanding the link between Borel Sets and Probability theory. How/Why are Borel Sets used in Probability theory?
2
votes
2answers
64 views

Evaluating $\int_0^\infty \frac{1}{(k-1)!} (\frac{x}{y})^{k+1} (1-y)^{-x/y} \, dx$

EDIT: I CHANGED THE QUESTION (I HAD THE WRONG BOUNDS!) THE ACTUAL QUESTION WAS FROM 0 TO INFINITY, NOT 0 TO 1! I'm stuck with evaluating this integral and I need some help! $$\large\int_0^\infty ...
1
vote
1answer
36 views

An Elementary Convergence Problem in Probability

Suppose that $X_1,X_2,...$ are degenerate random variables such that $f_{X_n}$ denotes the mass function of $X_n$.$$f_{X_n}(x)=P(X_n=x)= \begin{cases} 1, & x=2+\dfrac{1}{n} \\ 0, ...
2
votes
1answer
10 views

Conditional Probability of one RV having maximum among three

Let $X,Y,Z \sim \mathcal{N}(0,1)$ be independent. What is $\mathbb{P}(X>Y | Y>Z)$? I've come up with the following solution (is it correct?) but I cannot seem to understand it intuitively. ...
1
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0answers
9 views

When is the probability of countable union equal to the limit of probabilities of finite unions?

Lets say there are arbitrary sequence of sets $A_i$. When does the following below equation hold?, i.e., what specific properties of $A_i$ would make it invalid $$P\left(\lim_{n \to \infty} ...
0
votes
1answer
14 views

Simplifying this summation

I've been doing this question and I'm stuck! Each customer who enters Larry’s clothing store will, independently of every other customer, purchase a suit with probability p. Assume that N, the ...
1
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0answers
23 views

How to “reduce” a probability distribution satisfying certain conditions

I will try and explain the question I have in term of an example. I am given some probability distribution $f$, in this case of 2 variables x and p, $f(x,p)$. For example, I can pick the ...
0
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2answers
11 views

metrics for density-sampling similarity, beyond likelihood

I am looking for a metric that would evaluate the distance between a sample $S$ and a density function $D$ Building a sample from a known distribution can be done using a monte-carlo sampling, ...
1
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0answers
31 views

Identify a geometric theorem with probabilistic proof

Some months ago, I saw a theorem and its proof that was left on the blackboard from a previous computer graphics lecture. As far as I remember, the theorem went something like: It is possible to ...
-1
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0answers
19 views

Computation of probability from joint density function

Let $(X,Y)$ be a continuous rv with joint density $f(x,y) = k$ if $0<x<2, 0<y<1$ and $2y<x$, $f(x,y)=0 $ otherwise. I find that the constant $k$ is equal to $1$ and accumulated to ...
1
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2answers
28 views

Probability calculation of an event.

Suppose we have a village that has the following number of total rain days every year: A1, A2, A3, ...., An for n years. With Ax an integer number of course. We want to find the probability BASED on ...
0
votes
1answer
23 views

There are 2 red, 3 pink, 4 orange, and 5 yellow jelly beans in the pocket. how many different ways can you choose at least one jelly bean?

Thanks a lot!There are 2 red, 3 pink, 4 orange, and 5 yellow jelly beans in the pocket. how many different ways can you choose at least one jelly bean?
0
votes
1answer
30 views

Formula for X “successes” with X 10 sided die.

I am trying to create a formula for the % chance of having Y number of dice hit a number 8, 9 or 10 out of X possible. For example the chance of having 7 dice out of 10 dice be one of the 3 numbers. ...
0
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0answers
15 views

Measurability of function with two variables. [duplicate]

Let $\phi(x,v)$ be a function from $X \times \Theta$ to $\mathbb{R}$. Here $\Theta$ is an open subset of $\mathbb{R}^k$. And $\phi$ satisfies 1.Fix $x\in X$, $v\rightarrow \phi(x,v)$ is continuous, ...
0
votes
1answer
15 views

Finding the expectation of a random variable ($E[min(X,a)]$)

I will give some context first, then i'll ask the question. Suppose you have a random variable $Y = h(X)$, such that $ h(x) = \begin{cases} 0 & x < a \\ x-a & a \leq x < b\\ b-a ...
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votes
2answers
18 views

General approach for problems like “If a coin is tossed $n$ times, what is the probability that heads and tails appear $x$ and $y$ times”?

If a fair coin is tossed four times. What is the probability that two heads and two tails will result? I was solving the question above, since the sample space was small, I was able to list down ...
-1
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0answers
25 views

Correct Markov Model for Simple Exclusion Process

I'm given to build the Markov chain for Simple Symmetric Exclusion Process. Consider$C_n$, the circle of length n. Pick $n=3$. In each vertex can be a particle. In each step one pick a particle ...
1
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2answers
43 views

Combinatorics using a geometric diagram

How can I do this without trial-and-error? It has something to do with a triangle and summing the next row?
1
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0answers
19 views

Calculating variance and covariance of estimators. Where is the mistake?

I have a random variable $X$ and $N$ independent observation of it ($X_i, i\in\{1, \ldots, N\}$). We know that: $$\mathbb{E}[X_i^r] = \hat{\mu}_r,~ \mathbb{E}[(X_i - \hat{\mu}_1)^r] = \mu_r$$ I ...
0
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1answer
28 views

How can I solve this using permutations?

Delegates from 10 countries, including Russia, France, England and the United States are to be seated in a row. How many different seating arrangements are possible if the French and English delegates ...
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0answers
15 views

Book on Convergence Concepts in Probability without Measure Theory

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
0
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0answers
14 views

Sampling from overlapping domains

I would like to sample from overlapping domains and compute the expected number of domains sampled with every draw. As an example: Of the 20 students in Tanner's class, 8 wore a hat to school, 15 ...
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2answers
26 views

Convolution of 2 uniform random variables

I really do not know how to do this. Let $X$ have a uniform distribution on $(0,2)$ and let $Y$ be independent of $X$ with a uniform distribution over $(0,3)$. Determine the cumulative distribution ...
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0answers
11 views

probability question1201 [on hold]

A computer programmer averages one error per 60 program statements. Find the mean and standard deviations of the number of errors expected in a 250 statement program. Find the probability that a 100 ...
0
votes
1answer
27 views

Probability of getting an average of 3 or more by rolling 4 sided die twice

What I understood is the sample mean of two rolls of all sample space(16) as given below: ...
-1
votes
1answer
19 views

Probability on coin flips in n number of times [on hold]

this is a question from textbook which I was not able to solve please help (explain). Question: An unbiased coin is tossed "n" number of times. If the probability of getting 4 tails equals equals ...
1
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0answers
28 views

Probability of winning lottery

There are a total of $10,000$ tickets of which $500$ are winning tickets. Tim sells $200$ tickets. What is the probability that at least $12$ of them are winning tickets?
0
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0answers
11 views

What is a one-sigma Ellipse?

What is a 1-Signa ellipse? What does it represent? Also, I read the following sentence "One-Sigma lines of equal probability density of two normal distributions [(n^2 + n)/2 free parameters]" Why are ...