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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2answers
85 views

Does this paradox have a name?

As a student many years ago I learned of a paradox of something that is almost a certainty, while at the same time being highly improbable. For example, if you flip 10 coins the chances of all being ...
0
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1answer
41 views

What are random variables and its connection with functionals?

Here is an image of the conversation which I had with my Prof. (He's the one in violet and myself in orange) The topic was random variables and other probability related definitions. I tried to ...
1
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0answers
62 views

Probability:questions on characteristic functions

A well-known example to show that two random variables whose marginal distributions are normal, do not need necessarily be jointly normal is achieved by letting $X, Y $ have the following joint ...
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3answers
40 views

Confirmation on probability identity

I need a confirmation regarding a probability formula: $ P(A\cup B)=P(A\cap B)+P({ A }^{ c }\cap B)+P(A\cap { B }^{ c }) $ I am asked if this identity is true. For me, it is true. Can you please ...
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votes
2answers
38 views

Explain how assigning n tasks to n persons randomly gives a sample space of n^n?

I was studying linearity of expectation topic and there was an example of assigning n processes to n different servers randomly, as per i can think - 1st process to any of n servers so "n" ways, 2nd ...
-1
votes
3answers
68 views

Simulating a Six Sided Die with a Fair Coin [closed]

How can you you simulate rolling a die using the coin you have? Suppose you have the urge to play a game that requires tossing a biased coin with P(H)= 0.65 and P(T) = 0.35. How can you simulate ...
1
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0answers
33 views

Linear model for data that follow gaussian distribution

I have a question about linear regression. We have the linear regression of input data $(X,Y)=((x_1,y_1),(x_2,y_2)...(x_n,y_n))$ is $$F=aX+b$$ a,b are factors of the linear line, $y_i$ is {-1,1}. ...
1
vote
1answer
70 views

Expected value over many trials

I am a poker player and was talking to my friend about expected value. He claimed that if you play far enough above your bankroll, expected value can be negative, even if you have a skill edge. I ...
0
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1answer
45 views

Probability of identical twins

I have this problem statement: Suppose that in the population of twins, males and females are equally likely to occur and that the probability that twins are identical is $\alpha$. If twins are not ...
0
votes
1answer
17 views

Relationship between the probability two items in a list are correctly relatively ordered and the probability of any item being in the right place

Given a disordered list a of size n, and an ordering on the items of a, is there a relationship between the probability that two items are correctly ordered ($P(a_i \leq a_j)$ given $0 < i < j ...
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0answers
12 views

Combining independent Gaussian probabilities

I am using three Gaussian distributions with which I generate random numbers to represent many candidate xyz points. I use some selection criteria (details not particularly relevant) to decide on ...
-1
votes
2answers
58 views

One of the Actuary Exam P question.

For Company A there is 60% chance that no claim is made during the coming year. If one or more claim are made, the goal claim amount is normally distributed with mean 10,000 and standard deviation ...
1
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2answers
87 views

Accuracy of football prediction

Say you predicted that Brazil would have beaten Germany 2-1, by how large a percentage were you wrong? As Brazil scored 1 out of 2 goals, you could say that part of your prediction was 50% right. ...
0
votes
1answer
28 views

What are some other applied advanced probability sub-field relevant to finance?

What are some other applied advanced probability sub-field relevant to finance? I have heard Martingale, stochastic process, stochastic calculus, monte-carlo statistics I've been searching other ...
0
votes
1answer
68 views

Does $E[XY|Z]=E[XY]$?

Assume $E[X|Z]=E[X]$. Assume $Y$ and $Z$ are independent. Does $E[XY|Z]=E[XY]$? Can you prove it? My intuition says $E[XY|Z]=E[XY]$ but expanding the expectations into integrals I couldn't prove it. ...
1
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2answers
32 views

Need help understanding the difference between a.s. convergence and convergence in probability.

I have problem understanding the difference when I look at the alternative definition of a.s. convergence. I know how it is defined originally, but it is the alternative definition which makes it ...
2
votes
1answer
106 views

Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
0
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0answers
28 views

Countable Baye's theorem?

Disclaimer: If this is a foolish question, I'm sorry.. this is the first time I've looked at probability theory in very many years, and have begun to re-read everything from scratch... Question: If ...
-1
votes
1answer
44 views

Probability of 1 at both end of string

Given a string S having N characters long and consists of only 1s and 0s. Now given an integer K, let us pick two indexes i and j at random between 1 and N, both inclusive. What's the probability ...
0
votes
0answers
20 views

How to compute the integral of Gaussian Copula?

I want to compute the integral of Gaussian Copula in Matlab. The code is following: GaussianCopula = @(x,y) copulacdf('Gaussian',[x y],RHO); value = integral2(GaussianCopula,0,1,0,1) But,there are ...
-1
votes
1answer
23 views

Asymoptotic distribution of identically distributed random variables [closed]

$Y_1, Y_2, ..., Y_N$ 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 ...
0
votes
1answer
24 views

Distribution of the sum of two independent normal variables [duplicate]

Given the two variables $A\sim \text{N}(\mu, \phi^2)$ and $B\sim \text{N}(\xi, \omega^2)$ with $\mu, \xi \in R$ and $\phi^2, \omega^2 > 0$ how do I prove that $C := A + B\sim \text{N}(\mu + \xi, ...
3
votes
0answers
66 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 ...
2
votes
4answers
120 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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0answers
30 views

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

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
258 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
17 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. ...
3
votes
1answer
30 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 $S \in \Sigma_0$ $F\in \Sigma_0 \Rightarrow F^c \in \Sigma_0$ ...
4
votes
1answer
63 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
votes
1answer
58 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 ...
4
votes
1answer
94 views

Finding a specific sequence of digits in pi

Looking at the pifs project on GitHub and this question on SO has made me curious as to how feasible it is to find a specific sequence of digits within Pi. Essentially, on average, how many digits of ...
2
votes
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 ...
-1
votes
1answer
20 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 ...
0
votes
1answer
69 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 ...
11
votes
5answers
401 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
14 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 ...
2
votes
0answers
55 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 ...
0
votes
1answer
25 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 ...
0
votes
1answer
67 views

What will be probability of this problem? [closed]

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 ...
0
votes
1answer
23 views

Probabilty with profit question. [closed]

Probability => Profit First table 1/3 => 350 1/6 => 350 1/6 => -100 1/3 => -100 Second table ...
0
votes
1answer
55 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
0
votes
1answer
43 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 ...
0
votes
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 ...
0
votes
1answer
26 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: ...
1
vote
1answer
25 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 $(X_i)_{i\in\mathbb{N}}$ ...
3
votes
3answers
137 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 ...
1
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0answers
22 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. ...
1
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1answer
23 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 ...
0
votes
1answer
27 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) = ...
0
votes
1answer
44 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 ...