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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10 views

Joint distribution and the probability of event X>Y for discrete random variables X,Y

In a fourth year statistics course there are 10 actuarial science students, 9 statistics students and 6 math business students. Five students are selected at random without replacement. Let X be the ...
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1answer
9 views

Expression defined by exponential random variables, probability of being nonnegative

Consider $n \geq 2$. Let $E_1,...,E_n,F_1,...,F_n$ be independent exponentially distributed random variables with rate $1$. Define $T_E = \displaystyle \sum_{i=1}^{n}{E_i}$, and $T_F = \displaystyle ...
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2answers
26 views

Chance of winning simple dice game

Tossing two fair dice, if the sum is 7 or 11, then I win; if the sum is 2, 3 or 12, then I lose; if the sum is one of rest of numbers then I toss the two dices again. What is probability of winning? ...
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1answer
21 views

Probability of certain outcomes after moving a randomly chosen ball from bag A to bag B, and in the opposite direction

Bag A contains 6 yellow balls and 8 blue balls while bag B contains 5 yellow balls and 9 blue balls. A ball is picked from bag A and placed into bag B and then a ball is picked from bag B and returned ...
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1answer
35 views

Markov chain doesn't sum up to 1

Let $\{X_n\}$ be a Markov chain on $S=\{1,2,3,4,5,6\}$ with the matrix suppose we define a new sequence $\{Y_n\}$ by $$Y_n=\cases{1\quad X_n=1\vee X_n=2\\2\quad X_n=3\vee X_n=4\\3\quad ...
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0answers
20 views

Expected outcome for repeated dice rolls with dice fixing

Here is another dice roll question. The rules You start with n dice, and roll all of them. You select one or more dice and fix them, i.e. their value will not ...
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2answers
25 views

Calculating $E[X|Y]$ for continuous $X$ and discrete $Y$

I'm struggling with the following exercise, which I have the solution to but don't understand. I would appreciate any help. The exercise Let $X$ a random variable with $f_X(x) = 2x$ if $x \in [0,1]$ ...
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0answers
12 views

Have I used the Probability generating function of poisson point process correctly?

Let $v\in \mathcal{V}$ be measurable and let $\Phi$ be a Poisson Point Process with intensity $\lambda$ then the probability generating function (PGF) is $$\mathbb{E}\left( \prod_{x\in \Phi} ...
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1answer
23 views

An unusual “multivariate Gaussian integral” that comes up when trying to translate results about a standard Gaussian to the general case

I am trying to solve this question and it leads me to a strange looking integral that I do not know how to solve. Let $\Sigma$ be positive semidefinite, and $1>\lambda>0$. I am not certain I am ...
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1answer
30 views

Expected number of days of watching movie

Let us suppose I have a list of $n$ movies and each day I chose a subset of this movies where the probability of choosing any movie from the list is p . Now , what would be the expected number of days ...
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0answers
10 views

Expected size of largest connected component in a random k-out digraph?

Given a digraph with n vertices and kn edges, where each vertex has k out-neighbors randomly chosen at uniform without loops, how would I go about figuring out the expected value of the size of the ...
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3answers
29 views

Product of random independent variables

What are the properties of the product of random variables? The book I have on probability and statistics only comments on their sum properties. I know that when you sum random independent variables ...
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3answers
20 views

Determining number of combinations on odd dice

Suppose we had four dice with four sides. Two sides have a 1/10 chance of being rolled, two sides have a 4/10 chance of being rolled. The dice score points equal to 1, 3, 4, and 6, with the 1 and 6 ...
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1answer
16 views

How to interpret bracket, curly bracket, and the asterisk in Probability formula?

At the bottom of page #3 of this paper, it steates: $m_{jt}=\exp\{X_tB_j\}$ $X_t=[X_{1t} | X_{2j}] $ $X_{1t}=[1, x_{Djt} | X_t^*]$ May I know how should I interpret the bracket, curly bracket, ...
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1answer
17 views

How do you explain the strong sampling assumption

When using the strong sampling assumption, we assume that our data points are drawn uniformly and independently. In the example I recently saw we have a data set: $D = \{16, 8, 2, 64\}$. And we have ...
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1answer
26 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 ...
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0answers
36 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) = ...
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0answers
17 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 ...
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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 ...
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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 ...
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0answers
19 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 $a_n>0$ $\forall n$, $X_{n,j}= \frac{1}{a_n}I(|Z_j| \le a_n)$ ...
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2answers
23 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 ...
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2answers
28 views

Correlation between two variable

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
20 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 ...
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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 ...
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1answer
15 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!
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1answer
18 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 $$ ...
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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 ...
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2answers
47 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 ...
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3answers
22 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 ...
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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!
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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 ...
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6answers
73 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 ...
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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
13 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?
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2answers
62 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 ...
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1answer
29 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
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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. ...
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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} ...
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1answer
13 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 ...
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0answers
21 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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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0answers
16 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?
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1answer
29 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. ...