A probability provides a quantitative description of the likely occurrence of a particular event.

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14 views

Computing conditioned probability of X by $U=\min(X,Y)=u$

Lets $X$ and $Y$ be independent random variables with $P(X\leq x)=F_x(x)$ and $P(Y\leq y)=F_y(y)$. Lets $U=\min(X,Y)$. I know that $F_u(u)=1-(1-F_x(u)(1-F_y(u)))$ By definition: $P(X \leq x |U=u)= ...
2
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1answer
155 views

Birthday Problem: Difference between Canonical Solution and Approximation

A couple weeks back, I was seeing if I could solve the basic formulation of the Birthday Problem (i.e. assuming 365 equally likely birthdays, what's the probability that, given a room of ${n}$ people, ...
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0answers
12 views

Reference for inequalities involving limsup and liminf

What book contains proofs of those or contrapositives of those (either given or in the exercise sections)? I can't seem to find any in Williams' Probability w/ Martingales, Royden and Fitzpatrick's ...
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0answers
19 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf ...
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16 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
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0answers
18 views

Prove $P(T > \infty) = 0$

From Williams' Probability with Martingales How exactly do we know $P(T > \infty) = 0$? If the partial sums are bounded, then no matter how many terms in the sequence we add, they won't ...
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0answers
14 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
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0answers
29 views

Prove $|M_n^T| \le c + K$ [on hold]

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
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0answers
18 views

Prove $E[(M_{\infty} - M_n)^2] \color{red}{=} \sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2]$

From Williams' Probability with Martingales The last part, if I understand right means that $$E[(M_{\infty} - M_n)^2] \color{red}{=} \sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2]$$ given ...
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0answers
19 views

Clarifications on proof of Doob's Forward Convergence Theorem, warning related to it and proof of a corollary

From Williams' Probability with Martingales: $X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$ --> Is this supposed to be stronger than $\lim X_n$ does not exist because it's ...
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0answers
21 views

What kind of functions make the $\limsup$ proposition is true? What kind don't?

What kind of $f(n): \mathbb{N} \to \mathbb{N}$'s make the following statement true? What kind don't? $\limsup A_{f(n)} \subseteq \limsup A_n$ where $n \in \mathbb{N}$ (*) Well obviously the ...
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0answers
16 views

Show stopping time $T$ has finite expectation by finding an upper bound for $P(T=k)$

Given random variables $$X_1, X_2, \ldots \stackrel{iid}{\sim} P(X_i = 1) = p = 1 - q = 1 - P(X_i = -1)$$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n ...
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0answers
11 views

Existence of limit of upcrossings of a sequence of random variables

From Williams' Probability with Martingales: Is the corollary asserting existence of $\lim_N U_N[a,b]$? If so, how do we know it exists? Is it necessarily finite? If not, it is the ...
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1answer
30 views

sjPlot: probabilities. How to interpret?

I am running the following model in R: model = lmer(Tau ~ ageS*days+YrsOfEds*days+sex*days+tract*days + (1|SubjectID), data=long) With this model I am ...
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1answer
16 views

Why is multinomial variance different from covariance between the same two random variables?

We know that if $\big(X_1,X_2...X_k) \sim multinomial(n;p_1,p_2...p_k)$ then $X_i \sim bin(n;p_i) $ Then, $var(X_i) = np_i(1-p_i)$. But we have $cov(X_i,X_j) = -np_ip_j$. So doesnt that imply ...
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1answer
30 views

Logistic Regression Probabilities

I have performed a logistic regression on some data, and the function is: $p(x,y) = \frac{\exp(4.5 + 3.5x - 0.1y)}{1+\exp(4.5 + 3.5x - 0.1y)}$ This also gives you the probability of observing some ...
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0answers
19 views

Bivariate Poisson regression and Pythagorean Expectation for soccer(football) [on hold]

based on public data and using excel 2010 or after, I want to forecast/predict the football match winner.
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4answers
65 views

Weighted lottery

A charter school assigns seats by conducting a lottery. In the past, each child was given a number, and balls were selected randomly. This year, they are starting a new policy of giving low-income ...
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1answer
38 views

Exercise given a sequence of i.i.d. real RV and a distribution function $F$ need to verify three statements for $Z_n=n \min (X_1, \dots , X_n)$

Pardon the title, I couldn't think of a better one. The three statements below are to prepare for a tougher exercise (which I could solve) however I am lost on 1 of the statements. Problem: Let ...
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1answer
48 views

Looking for a distribution with very specific properties

I'm looking for a continuous distribution which I can parameterize such that The expected value is roughly zero The expected maximum given $x$ draws from that distribution is only very weakly ...
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0answers
10 views

Assign a probability to a member of a group to be in another group [on hold]

I have 3 groups (A, B, C) of data and each group can be divided in nA, nB, nC subgroups. For each group I have then computed the percentage of subgroups that meet a certain property (pA,pB,pC). What ...
2
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1answer
19 views

Evaluating two sets of random samples

Let $p$ be a probability distribution that can be computed tractably for any given point. I use two MCMC methods to generate samples from the distributions. For each MCMC method, I run 1000 Markov ...
3
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1answer
321 views

Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} ...
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1answer
39 views

Proving the probability integral transform without assuming that the CDF is strictly increasing

I know that the proof of the probability integral transform has been given multiple times on this site. However, the proofs I found use the hypothesis that the CDF $F_X(x)$ is strictly increasing ...
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2answers
46 views

What am I doing wrong in this Baye's theorem equation

I have to find the probability that that a woman does not have cancer, given that her mammogram came back positive. Facts needed: 0.1% of all women have breast cancer probability of a positive ...
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1answer
28 views

Questions on Examples of Introduction to Probability Models

Here is the question: Suppose there are 25 different types of coupons and suppose that each time one obtains a coupon, it is equally likely to be any one of the 25 types. Compute the expected number ...
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0answers
12 views

Probability of descendants [on hold]

I am looking for a routine to estimate the probability that a person will not have descendents in the xth following generation. I assume that one would start with a probability distribution of the ...
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0answers
15 views

Probabilities of different length and expressed genes

So let's say I have 3 different genes that I am sequencing (this means I will get a lot of fragments of each and by overlapping I can get a complete gene) them. Each gene has a different length and ...
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0answers
24 views

How to interpret daily vs. hourly “% chance of rain” precipitation forecasts?

Looking at the National Weather Service forecast for daytime precipitation tomorrow, it shows a 70% chance of rain ("showers likely"). I interpret this as meaning that on average it will rain 7 out of ...
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0answers
52 views

Exercise on Borel Cantelli Lemma ($\lim \sup X_n/ \ln(n) =1$ a.s.) help required to rigorously write the statement

I hope this question is within the scope of this site. Please note that I have solved this Exercise, I do have doubts about my presentation though and about how to rigorously empathize on the ...
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0answers
30 views

Probability question regarding independence of events

The question was posed in this way: Suppose John and Jan read a book separately. Each person has a probability of $.5$ to catch a typo and the each event to catch a typo is independent. Let $A = ...
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0answers
39 views

Dependent/Conditional Probability over time [on hold]

I deal with statistics to quantify and describe time series all the time, however, I was asked by a friend to help provide a solution to his problem. The scenario is essentially this: 100 empty ...
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1answer
35 views

Why is it necessary to fix a matrix diagonal and after this calculate the exponential to assess transition probabilities?

I'm learning markov chains in order to compute estimations of transition probabilities, and I found an example of the estimator construction for continuous time markov chains: ...
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1answer
34 views

How to do calculate both causal and diagnostic inferences simultaneosly in bayesian networks?

Consider a simple Bayesian network as given below. Question: How to find $P(S|C,W)$? It is fairly straight forward to compute the causal inference $ P(W|S) = P(W|S,R)\cdot P(R) + ...
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0answers
20 views

Estimator of true probability — understanding margin of error for very small probability

I have a coin whose probability of landing on heads when flipped is unknown, but could be anywhere between 0 and 100%. I want to flip the coin some number of times and estimate the true probability ...
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0answers
8 views

Q: Probability of category k yielding the maximum occurrence count after N IID trials

Assume we have a categorical distribution for random variable $X$ with $M$ categories. We generate $N$ IID realizations of the random variable and count the occurrences of each category in the sample, ...
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0answers
16 views

Zero-inflated Poisson (ZIP) [closed]

I am really struggling with this question. Could anyone please shed some light?
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0answers
44 views

If the # of people in a room is Poisson distributed, and you observe someone enter, what's the distribution of the # of people?

I hope this question is properly formulated. It's just something that occurred to me spontaneously. Consider a random variable $X$ representing some count data -- for instance, the number of people ...
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0answers
11 views

Probabilty to choose x times (or more) one bin with a specific probabilty

I've two bins, one bin (bin A) has a probability y to be taken, the other (bin B) 1-y. I pick N=1000 times one of the two bins (using their respective probabilities). What is the probabilty to choose ...
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0answers
15 views

posterior distribution for coin flips given uniform prior distribution [closed]

A coin has an unknown head probability $p$. Flip $n$ times, and observe $X=k$ heads. Assuming an uniform prior for $p$, then the posterior distribution of $p$ is the beta distribution $B(\alpha = k + ...
1
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1answer
43 views

transformation of uniform distribution variable

Problem is this. Given probability density function of $ f(x)=1 , \phantom2 0<x<1 $ when variable $X$ is transformed into $ Y=-\log(X) $, I have to find the probability density function of $Y$ ...
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17 views

probability of non-exceedance or the probability of exceedance for weibull distribution [closed]

How to calculate the probability of non-exceedance or the probability of exceedance in R for Weibull distribution? R package "nsRFA" is available in order to calculate probability of non exceedance ...
3
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1answer
78 views

Simple example wanted: $ X_n $ converges to $X$ in probability but not almost surely

I'm looking for a simple example sequence $\{X_n\}$ that converges in probability but not almost surely. The example I have right now is Exercise 47 (1.116) from Shao: $ X_n(w) = \begin{cases}1 ...
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1answer
23 views

Intuitively, why is cross entropy a measure of distance of two probability distributions?

For two discrete distributions $p$ and $q$, cross entropy is defined as $$H(p,q)=-\sum_x p(x)\log q(x).$$ I wonder why this would be an intuitive measure of distance between two probability ...
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3answers
211 views

Simulate a Bernoulli variable with probability ${a\over b}$ using a biased coin

Can someone tell me how to simulate $\mathrm{Bernoulli}\left({a\over b}\right)$, where $a,b\in \mathbb{N}$, using a coin toss (as many times as you require) with $P(H)=p$ ? I was thinking of using ...
2
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2answers
63 views

What is the difference between $1 - p^k$ and $(1-p)^k$?

Let $p$ be the probability that an event happens, then $1-p$ is the probability of that event not happening. The probability of that event happening $k$ times (in a row), i.e. rolling the same number ...
2
votes
1answer
56 views

Expected time to get all four unique coupons [duplicate]

Envelopes are on sale for Rs. 30 each. Each envelope contains exactly one coupon, which can be one of four types with equal probability. Suppose you keep on buying envelopes and stop when you collect ...
3
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1answer
35 views

Proof of Set-theoretic limits

I'm trying to solve this problem but there is something unsure about my solution. Given $lim_{n \rightarrow \infty}supA = \cap_{n=1}^{\infty} \cup_{k=n}^{\infty}A_{k} $ $lim_{n \rightarrow ...
2
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0answers
41 views

Exercise in computing $P(X>0,Y>0)$ given a distribution density function for $(X,Y)$

Exercise: Given $p \in (0,1)$ and $(X,Y)$ a pair of RV with distribution density function $$f(x,y)= \frac{1}{2 \pi \sqrt{1-p^2}}\exp \left( -\frac{1}{2(1-p^2)}(x^2-2pxy+y^2) \right) $$ Compute ...
4
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4answers
79 views

Odds of a number appearing only one time in 500 spins on roulette

I recently observed a roulette wheel where one number appeared only once in five hundred spins. This is an American roulette wheel with 0 and 00, so the odds of any number hitting should be 1/38. I ...