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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2
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1answer
37 views

determine the distribution of the random variable $Y=\Sigma_{k=1}^{\infty}kX_k$

Fix $p \in (0,1)$ and consider independent Poisson random variables $X_k$, $k \geq 1$ with $\mathbb E[X_k]=\frac{p^k}{k}$. Verify that the sum $\Sigma_{k=1}^{\infty}kX_k$ converges with probability ...
3
votes
1answer
18 views

probability question with marbles

I have $13$ different color marbles. One color is $5$ times likely to be chosen and another color is half as likely to be chosen. What is the probability that, $1$. you choose the marble that is 5 ...
1
vote
0answers
18 views

Multivariate normal distribution conditional probability question.

$\newcommand{\Cov}{\operatorname{Cov}}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\E}{\mathbb{E}}$$\newcommand{\P}{\mathbb{P}}$We have that $X$ and $Y$ are random variables with a ...
0
votes
0answers
27 views

Proving that an integral of several cdf and pdf functions is increasing in a certain parameter.

Basic assumptions: $n\geq3$, $a\leq b\leq c$, $b$ is simply a dummy variable of integration, and $\rho\geq0$. $F(z)$ and $f(z)$ represent the usual general CDF and PDF (no specified distribution here)....
4
votes
1answer
51 views

conditional probability on zero probability events and conditional Radon-Nikodym derivatives

Consider a stochastic process $\{x_t\}_{t\in T}$ adapted to some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}\}_{t\in T},\mathbb{P})$ taking values in the state space $(\mathbb{R},\...
0
votes
1answer
34 views

How many different groups of 12 people can be chosen from a group of 30. More restrictions on details.

How many different groups of 12 people can be chosen from a group of 30. Note: the group of 30 contains: 2 people that will not work together (pick neither, or pick one, but not both) and 2 people ...
1
vote
1answer
52 views

Expectation of $|H - T|$

Using binomial approximation to normal distribution, find the expectation of $|H-T|$ where the $H,T$ are heads and tails of a fair coin and the number of tosses is large. Can anyone please tell me, ...
2
votes
1answer
56 views

probability of rank of a number

Suppose I have 10 sample means. I want to find the probability of rank of the population means using sample means. Therefore, I want to perform two experiments. First experiment: I pick one of the ...
2
votes
1answer
36 views

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to?

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to- $\frac{\Phi(1)-\frac{1}{2}}{\Phi(2)-\frac{1}{2}}$ $\frac{\Phi(1)+\frac{1}{2}}{\Phi(2)+\frac{1}{2}}$...
3
votes
2answers
56 views

How to precisely define a function that chooses randomly from a finite set?

Let $A = \{1, 2, \ldots, n\}$. I want to define a function that picks with uniform probability an element in $A$, so that $$f(A) = i \in A.$$ I don't know how to precisely define this mathematical ...
0
votes
3answers
29 views

Does $1-\mathbb{P}(X_1>x_1, X_2>x_2)=\mathbb{P}(X_1\leq x_1,X_2>x_2)$ hold?

I am wondering does $1-\mathbb{P}(X_1>x_1, X_2>x_2)=\mathbb{P}(X_1\leq x_1,X_2>x_2)$? Even if $X_1$ and $X_2$ are dependent?
0
votes
1answer
16 views

Getting the joint function. What am i doing wrong?!?

we have that $f(x_1,x_2)=2(1-x_1)$ if $0≤x_1≤1$, $0≤x_2≤1$. And we have that $Y_1=x_1x_2$ and $y_2=x_1$ And i have to find the joint distribution of $y_1$, $y_2$:(f($y_1,$$y_2$)) and verify if this a ...
1
vote
1answer
20 views

conditional probability (discrete case).

I am not sure if I am doing this right.  We have this table $$\begin{array}{r:r|rr} & & X\\\hdashline & & 1 & 2\\ \hline & 0&.12 & .08\...
1
vote
1answer
55 views

Calculate the probability that the running total is exactly n. (homework help)

I am working through Harvard's public Stat 110 (probability) course. Question: A fair die is rolled repeatedly, and a running total is kept (which is, at each time, the total of all the rolls up ...
-3
votes
0answers
60 views

planning on trading, need mathematical edge [on hold]

I have been looking in to binary options trading, How It Works Retail trader (maybe me) goes to broker to trade binary options. If I trade that I think Euro/USD currency pair will go down, then I ...
1
vote
1answer
62 views

What is the probability of two-pair poker hand?

To start with, this question has never been asked as how I am going to ask: What is the probability that a five card poker hand will have two pairs (with no additional cards)? Example of two-...
1
vote
0answers
35 views

Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
0
votes
0answers
15 views

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$ $o_i$ = observed $e_i$=expected (I removed $i$'s for ease) The solution is: $$G^2=2\sum o \log \frac{o}{e}$$ $$=2\...
0
votes
1answer
40 views

Coin toss related problem

What is the minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least 0.96? Is there any shortcut way to calculate this?
2
votes
0answers
33 views

What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
0
votes
1answer
52 views

Show that $\frac{S_n}{n}\to 0$ in probability if $s<\frac{1}{2}$

Let $s\in\mathbb{R}$ and $X_1,X_2,\dots$ be independent random variables and with distributions: $$P(X_n=n^s)=P(X_n=-n^s)=\frac{1}{2}$$ Let $S_n=X_1+\dots+X_n$. Show that $$\frac{S_n}{n}\to 0 \text{ ...
0
votes
1answer
29 views

Integration by parts vs expanding giving different answers.

You are given the probability density function of a random variable X. $$f_X(x) = 2x$$ $$0<x<1$$ Find the difference between the third central moment and the second central moment of this ...
0
votes
1answer
29 views

Probability of choosing a number from the set $\{1,2,\ldots,99\}$ that divided by $5$ has the remainder $2$ and is a multiple of $3$

Good evening to everyone. I have to find the probability of choosing a number from the set {1,2...99} that divided by 5 has the remainder 2 and at the same time it's multiple of 3. I know that the ...
2
votes
1answer
45 views

Calculating the number of “birthday days” in the birthday problem

Given 's' students in a room and 'd' days in the calendar year, what is the probability 'P' that there will be 'k' "birthday days"? i.e., 'k = 1' means that everybody's birthday falls on the same day,...
0
votes
0answers
19 views

Convolution of multiple exponential distributions

I'm trying to figure out the derivation presented on page 442 of this paper. Given a probability distribution $$f_n(t) \frac{\binom{n+1}{2}}{2N}\exp{\left(-\frac{t\binom{n+1}{2}}{2N}\right)}$$ ...
0
votes
2answers
31 views

basic Quantile proves

Let this be my definition of a quantile funktion. X is a real-valued random variable. And let F be it's distribution function. then \begin{align*} F^{-1}(a):=\inf\{x\in \mathbb{R}: F(x) \ge a\}. \end{...
1
vote
1answer
50 views

Factories processing jobs

We have two factories that can process jobs; each job takes two days to complete. The factories agree on a minimum threshold $a\in[0,1]$ to accept jobs. Every day, a value $v\in[0,1]$ is drawn ...
0
votes
3answers
30 views

How to find E(x) and Var(x) in this specific continuous probability distribution.

I've got into some confusion on continuous probability distributions, and everything related to it. This is the problem: Problem Image. I assume from the sketch that pdf is $f(x) = x$ for values of $x$...
0
votes
0answers
27 views

What would be a basis of $L^2(\Omega )$

Let $(\Omega ,\mathcal F,P)$ a probability space and $$L^2(\Omega )=\{random\ variable\ X\mid \mathbb E[X^2]<\infty \}$$ is a vector space. What would be a basis of $L^2(\Omega )$ ? I also know ...
2
votes
0answers
36 views
+50

Expected score from threshold with number deletions

We play a game where a sequence of $n$ numbers is drawn uniformly from $[0,1]$, and we need to set a threshold $0\leq a\leq 1$. For every number that is at least our threshold, we get $a$ points but ...
1
vote
1answer
49 views

Proof that $ \sum_{i=1}^\infty a_n$ is converges almost surely.

Let $\{a_n\} $ be a positive number sequence and $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. If we have $$\sup\limits_{t>0} \left( t. \mathbb{P} \left\{ \sum_{i=1}^\infty a_n >t \...
0
votes
0answers
11 views

CGF determines the distribution

It is well known, that if the domain of the mgf of a random variable $X$ contains an interval around zero, then the distribution is completely determined by its moments. However consider a Levy-...
0
votes
0answers
19 views

Event ($E_n$, i.o.): an exercise for understanding the definition

This is an exercise from David Williams' "Probability with Martingales" on page 28. I am studying on my own and I would appreciate some feedback on my proof. For an event $E$ define an indicator ...
0
votes
0answers
18 views

estimation for analytic stochastic processes

this is for experts in probability and stats. There is a theorem, I have seen once: Given a stationary analytic random process, one can show that from the values of a sample path in a finite interval ...
0
votes
1answer
22 views

Ornstein-Ulhenbeck and convergence almost surely

Let this O-U equation : $$ dX_t = \alpha X_t dt + \sigma dB_t,\ \ \ X_0=x $$ where $x,\alpha<0,\sigma$ are constants. I proved that $X_t\xrightarrow{d} X\overset{d}{=}\mathcal N (0, \frac{\sigma^2}...
1
vote
0answers
35 views

Probability question, can I reset the window or not

There is a wall street banker. The banker invests in a kind of share called as options. The main features of this share is as follows: You make a bet with a specified amount of information as to ...
1
vote
1answer
22 views

Let $X$ be the number of trials until you roll two ones. Derive a formula for $P(X > x)$

Suppose $X \sim \textrm{Geom}(p)$ on $\{1, 2,\dotsc\}$. Roll a pair of dice until you get two ones on the same roll. Let $X$ be the number of trials when you stop. Derive a formula for $P(X > x)$ ...
0
votes
1answer
26 views

Probability density function of Poisson Process trajectory

Given a Poisson Process with rate $\lambda$, by a fixed time $t$ we have observed $n$ arrivals at times $t_1 < \cdots < t_n$, with $t_0 = 0 < t_1$ and $t_n < t$ I'm trying to find a ...
0
votes
0answers
19 views

Constraints on Mutual Information Independence Test

Suppose all variables are binary for the sake of simplicity. There is a theorem about mutual information (MI) and a distribution $\chi^2$. Given a data set D with N data points, if the hypothesis ...
0
votes
1answer
19 views

Having trouble calculating expected value?

In a mall, a survey found that the number of people who pass by JCPenney between 4:00 and 5:00 pm is a Poisson random variable with parameter λ = 100. Assume that each person may enter the store, ...
0
votes
1answer
20 views

p(a,c) vs p(a∧c)

In this paper: https://www.aclweb.org/anthology/J/J16/J16-2006.pdf, the author breaks the Pointwise Mutual Information of a phrase up into several components: They use the ...
0
votes
1answer
21 views

How to model guessing?

I want to model the knowledge of the student $i$, in a particular subject $S$. I give him a set of questions $Q$ from $S$ to test his knowledge. The level of his knowledge depends on the number of ...
2
votes
1answer
46 views

I stumbled on a relationship between ln(x) and estimated probability. Can someone help me locate or generate a proof?

Yesterday, I personally stumbled on the following relationship of ln(x): Say you have x number of checkboxes, and you randomly pick a position (p) between 1 and x. If the checkbox at position P is ...
1
vote
1answer
42 views

Inequality: product of integrals

Context: Proving integral inequalities about posterior distributions following different sequences of binary signals. The proofs come down to the following inequalities. Let $\psi(x)$ be a concave ...
1
vote
1answer
31 views

Solving a recurrence relation of conditional probability functions

Suppose you have the recurrence relation for a probability function Q: $$Q(n_1,n_2|n) = Q(n_1-1,n_2|n-1)\frac{n_1-1}{n-1} + Q(n_1,n_2-1|n-1)\frac{n_2-1}{n-1}$$ where $n = n_1 + n_2$ and the ...
1
vote
1answer
40 views

What is the probability that a majority vote gives the correct answer, given that we know the accuracy of each of the voters?

Let's say that we have 7 voters who are voting on a decision. Furthermore, we know that voter A makes the right decision with 10% probability. voter B makes the right decision with 20% probability. ...
0
votes
1answer
31 views

Probability Question: Mutually Exclusive Events

Source of original question and answers can be found here under "Exercise 1" http://www.intmath.com/counting-probability/9-mutually-exclusive-events.php A box contains 100 items of which 4 are ...
0
votes
2answers
21 views

Does this hold in every case, and if only this one, why? Expectation, mean of random variable.

Characteristic function of random variable $X$ let us denote as $f_X(t)$ and $EX$ it's mean or expectation. Does the following hold in all cases, because it keeps coming up and I don't know why it is ...
6
votes
3answers
64 views

What is the probability that $7$ cards are chosen and no suit is missing?

Cards are drawn one by one from a regular deck ($13$ cards for each of the $4$ suits). If $7$ cards are drawn, what is the probability that no suit will be missing? Ok, so I tried the approach ...
1
vote
3answers
41 views

Identical Obects in Permutation and combination

There are $2$ identical white balls , $3$ identical red balls and $4$ green balls of different shades. The number of ways in which the balls can be arranged in a row so that at least one ball is ...