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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Asymoptotic distribution of identically distributed random variables

$Y_1, Y_2, ..., Y_N$ are independent and identically distributed random variables with the distribution function $F := F_{Y_1}$ and $F'_n(y) = \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{\{Y_i \leq x\}}$ as ...
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1answer
10 views

Independence of sum two independent variables

Given the two variables $A$ ~ $\text{N}(\mu, \phi^2)$ and $B$ ~ $\text{N}(\xi, \omega^2)$ with $\mu, \xi \in R$ and $\phi^2, \omega^2 > 0$ how do I prove that $C := A + B$ ~ $\text{N}(\mu + \xi, ...
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0answers
23 views

Minimal number of edges removed to make a graph triangle free

I'm interested in finding an upper bound on the expected value of the minimal number of edges one needs to remove from a random graph $G_{n,p}$ (where each edge appears with probability $p$) in order ...
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4answers
61 views

Is it true that $\mathbb E[{\frac{X}{Y}]}={\frac{\mathbb E[X]}{\mathbb E[Y]}}$?

If $X$ and $Y$ are both random variables, does it hold $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
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1answer
12 views

What is the intuition behind the word “irreducible” as used in Markov chains?

The definition of the word irreducible for a Markov process is that it is a subset of our state space with the condition that all states within the subset are in communication with one another. I am ...
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2answers
232 views

Why is a random variable called so despite being a function?

According to my knowledge, its a function $P(X)$ which includes all the possible outcomes a random event.
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0answers
10 views

Conditional Mutual Information of Markov Chains

I'm attempting to use conditional mutual information to determine the order of a Markov chain, but having trouble relating notation used in a specific reference on this topic to the actual quantities. ...
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1answer
28 views

Terminology on algebra.

In Probability textbook, algebra usually defined as follows: A collection $\Sigma_0$ of subsets of $S$ is called an algebra on $S$ if $S \in \Sigma_0$ $F\in \Sigma_0 \Rightarrow F^c \in \Sigma_0$ ...
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0answers
25 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
2
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1answer
30 views

Count 1-bit in binary integers

Given an integer range [A,B], (1) What’s the probability to get a 1-bit if we first randomly choose a number x in the range and then randomly choose a bit from x? (2) What’s the expected number of ...
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2answers
18 views

Probability distribution of selecting combinations of green and yellow balls from a set of green/yellow/red

Let's say I have G green balls, Y yellow balls and R red balls. I'm interested in ...
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1answer
14 views

How to compute the bivariate normal distribution CDF with rho in Matlab?

I want to compute the bivariate normal distribution CDF with rho in matlab, but I just find mvncdf function http://www.mathworks.com/help/stats/mvncdf.html. I do not know how to use it and how to use ...
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0answers
21 views

How to compute the double integral of Gaussion copula in matlab?

I want to compute the following expression: The expression of Gaussion copula can be found here: How to compute the seconds-order partial derivatives of Gaussion copula? When I was trying to ...
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4answers
351 views

The Price is Right optimal play

The following situation happened on the Price is Right and I was curious about the optimal response. The rules are: A contestant rolls a wheel with 5 cent increments from 5 - 100 (20 numbers total). A ...
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0answers
11 views

Spherical Sampling of Projected Disk

Given a 2D solid disk centered at some 3D point $\vec{x}$ with radius $r$ and normal $\vec{N}$, I need to compute a random unit vector from the origin that hits the disk. The vector needs to be ...
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0answers
27 views

How is conditional probability being used here?

Because of conditional probability: $P(A\mid B)=P(A,B)/P(B)$, $$P(C(t)\in dt\mid x(T^+_{i-1}),x(T^-_{i}))=\dfrac{P(C(t)\in dt,x(T^-_{i})\in dx\mid x(T^+_{i-1}))}{P(x(T^-_{i})\in dx\mid ...
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1answer
14 views

Expectation of excess demand

Suppose random variable D has C.D.F. F. D is demand and y is supply in this case. Now, excess demand (D-y), D>y is lost and excess inventory (y-D), y>D is wasted. I have to find Expectation of lost ...
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1answer
49 views

What will be probability of this problem?

given a string S. It is N characters long and consists of only 1s and 0s. Now Given an integer K, we have to pick two indices i and j at random between 1 and N, both inclusive. What's the probability ...
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1answer
22 views

Probabilty with profit question. [on hold]

Probability => Profit First table 1/3 => 350 1/6 => 350 1/6 => -100 1/3 => -100 Second table ...
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1answer
35 views

How to compute the seconds-order partial derivatives of Gaussion copula?

How to compute the following seconds-order partial derivatives of u and v for Gaussion copula?Thanks
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1answer
23 views

I want to use the first-order and seconds-order partial derivatives of t copula in matlab, how to find their formulas?

I want to use the first-order and seconds-order partial derivatives of t copula in matlab, however, I cannot use diff() function to get its first-order and seconds-order partial derivatives. Who knows ...
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0answers
34 views

Simple Variance approximation I don't get

I have $$ \log(\lambda_j -1) = c + \alpha_{j-1} $$ From here I know $$ \lambda_j -1 = \exp(c+\alpha_{j-1}) $$ Then, they say $$ Var(\hat \lambda_j)\approx\ Var(\hat \lambda_j - 1) \approx ...
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1answer
20 views

Limits for expected value in a proof

I have a small step in a proof, that I'm not sure if I got it right. We have given the function $f(s):=\mathbb{E}[e^{\lambda S (s-1)}]$ where $S$ is a random variable such that: ...
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1answer
24 views

Showing that a set is in terminal $\sigma$-Algebra

I am reading a probability theory book (from Bauer) and I found the following statement in the book that I cant understand: Given a sequence of independent random variables $(X_i)_{i\in\mathbb{N}}$ ...
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3answers
125 views

What's the probability a random number is at least twice as big as another?

Two numbers $m,n$ are chosen from a normal distribution, i.e. the chance that either number lies between $a$ and $b$ is $$\frac{1}{\sqrt{2\pi}}\int_a^be^{-x^2}dx$$ Edit: you could also say ...
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0answers
18 views

How to compute cumulative intensity process integral?

I am faced with a basic question about counting process and its intensity process used in survival analysis. It is actually the textbook example from Aalen's Survival and Event history analysis book. ...
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1answer
22 views

Probability of no more than X events in Y days (Poisson distribution)

I have to calculate the probability of no more than 8 events happening in 3 days, given the rate (5). I know I should calculate p(x+y+z)=p(x)*p(y)*p(z), where x+y+z<=8, but as there are quite many ...
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1answer
21 views

Binomial distribution or probability intersection

I flip a biased coin, p = 0.5 for getting heads. What is the probability of getting heads 8 times ? Firstly I used probability intersection $$ P(A \cap B \cap C \cap D \cap E \cap F \cap G \cap H) = ...
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1answer
24 views

Orthogonal transformation of multivariate normal

Let $X \sim N_n(\boldsymbol{\mu}, I )$. Let $O$ be an orthogonal matrix, with the first line $\frac{\boldsymbol{\mu}^T}{\|\boldsymbol{\mu}\|}$, and $Y=OX$. It can be proved that ...
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0answers
9 views

Non centered Chi-Squared distribution

Let $X \sim N_n(\boldsymbol{\mu}, I )$. Let $O$ be an orthogonal matrix, with the first line $\frac{\boldsymbol{\mu}^T}{\|\boldsymbol{\mu}\|}$, and $Y=OX$. It can be proved that ...
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0answers
21 views

Distributions with infinity variance.

I'm looking for a list (or something like that) of distributions with infinity variance (or infinity second moment), like non-gaussian Stable Distributions. I have an important warning: Some ...
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1answer
42 views

I need help understanding this proof about convergence in distribution

The proof says that we used the fact that $(1-\epsilon)^\frac{x}{\epsilon} \rightarrow e^{-x}$ Why is this so? How do I prove this? Also, why do we need the fact that $\lfloor x/p_n \rfloor - ...
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0answers
23 views

differential equation with random coefficient

I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself: Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < ...
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2answers
36 views

Finding mean from die probability

Example 4.4.5: Suppose that there is a 6-sided die that is weighted in such a way that each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 5 are all equal, ...
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72 views

Number of ways to get 4 consecutive numbers out of 5 dice? [on hold]

Suppose 5 fair dice are rolled. Consider the dice as indistinguishable. How many different outcomes produce a sequence of 4 consecutive numbers? A. 2 B. 12 C. 96 D. 11 E. 3 How many ways are there ...
3
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1answer
35 views

Upper Bound on Mutual Information

I am interested in an upper bound on mutual information that I have been encountering frequently in the statistics and probability literature. I have yet to see the "purest" form of the inequality, so ...
2
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1answer
22 views

Marginalization of a paramter in Gaussian

If $\theta \sim N(\mu,\sigma_o^2)$ and $\mu \sim N(0, \sigma_1^2)$ what is the marginalized $P(\theta)$. Is it $N(0,\sigma_o^2+\sigma_1^2)$?
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4answers
357 views

Probablity that 3 husbands sit next to their wives round a circular table

There are 3 couples sitting randomly round a 6-seater circular table. What is the probability that all the husbands and wives sit next to each other? My attempt: First wife, say, takes any of the ...
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0answers
12 views

Error of a Serial Processs

Give random variable X and two processes A, B . Assume that $ Y_{1}, Y_{2}$ are estimated versions of X by using processes A, B respectively, with probability: $P\left \{ \left | X-Y_{1} \right ...
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1answer
59 views

Given a person's age in years, what is the best way to estimate their age correctly in the future?

I have a database of people who have only given their age in years (and the date that they specified it). Because I want to keep this number accurate over time, I want to convert ages to birth years. ...
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1answer
15 views

Deriving marginal effects in multinomial logit model

For the multinomial logit model, it holds that: $$P[y_i=j]=\frac{\exp{\beta_{0,j} + \beta_1 x_{ij}}}{\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})}$$. Now my book states that the marginal effect is as ...
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2answers
30 views

Application of Holder's Inequality

Suppose $X$ has finite variance, $Y$ is normally distributed. $X$, $Y$ may not be independent. We denote $Z = XY$ . Show that $E|Z|^n < \infty$ for all $1 \le n < 2$; I tried writing down ...
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1answer
19 views

Calculating percentage of better hand.

For example, I have two cards and there are two other players. They have both 20% chance of having a better hand then me, how much chance is it in total that either one of them has a better hand then ...
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1answer
17 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim Bin(n_i, p_i)$. We know that $$Pr[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is there an easy way to ...
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1answer
43 views

Probability distribution of the third side in triangle

Given the two distributions of two sides of a triangle (for example, Uniform and Rayleigh) and the distribution of an angle between them (Uniform[0,Pi]), find the length of the third side. What i ...
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0answers
17 views

When do almost all random variables attain the expectation?

Given some sample space $\Omega$ I choose uniformly at random some $X \in \Omega$. Assume that I know the expected value $\mathbb{E}(X)$. What are the further conditions I need if I want to talk ...
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0answers
17 views

measure of dependence for copula

I have some question about the paper of Schweizer and Wolff (1981). The question concerns about the following bound $$\int_0^1\int_0^1|C(u,v)-uv|\,du\,dv\leq\frac{1}{12}$$ where $C$ is any copula. ...
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0answers
35 views

Intuition in probability theory

Good afternoon. Could you please suggest me some books or may be articles where I can read about the intuition of Kolmogorov's axiomatics. I know it, I can solve university problems but I can't feel ...
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0answers
15 views
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0answers
14 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...