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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0
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1answer
16 views

Find the probability of opening all the boxes

Suppose there are 20 boxes which 1-20 are printed on each box. There is a key in each box which are also marked with 1-20. So only the key with the same number with the box can open it. For example, ...
0
votes
1answer
10 views

When do I use Law of total variance?

For example, at the beginning of doing this problem (http://math.illinoisstate.edu/krzysio/3-6-10-KO-Exercise.pdf), I was thinking of using $\text{Var}(\text{Total loss}) = \text{Var}(N \cdot L)$, ...
0
votes
1answer
13 views

how to find the cumulative density function

Consider $$f(x)=3x^{-4} \qquad \mbox{on} \qquad x\geq 1.$$ Let $X$ be a continuous random variable on $x\geq 1$. Find the cumulative distribution $F(x)$ for $X$. I know that CDF for a continuous ...
6
votes
5answers
251 views

If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, ...
0
votes
0answers
16 views

Assume a die is rolled repeatedly. Find the markov matrix $P$ for the random variable of the time until the next $6$.

Assume a die is rolled repeatedly. Find the markov/transition matrix $P$ for the random variable $X_r$ = the time until the next six at time $r$. My solution was: For $i,j \geq 0$, $P$ is given ...
1
vote
3answers
20 views

The non-uniform probability of sums from the throw of multiple dice

I'm reading The Drunkards Walk by Leonard Mlodinow. In the book, the author writes: From a throw of three dice, a sum of 9 and 10 can be constructed in an equal combinations. However, the outcome ...
0
votes
0answers
42 views

Probability of choosing a point from infinite set

Let x and y be non-negative integers and $y \le x \le m$. Let us define a function $ f(x) = x/n, n = 1,2,3,... $ For a value $ m $, what is the probability of selecting a point $ p(m,y) $ so that $ ...
1
vote
0answers
30 views

Application of Slutsky's Theorem to the Convergence of Sum of R.V.

Let $X_1, X_2,…, X_n$ be i.i.d. $U(−\theta,\theta)$. Show that $Z_n \to N(0,\sqrt{\frac{5}{9}}$ in distribution, where $Z_n ...
0
votes
0answers
22 views

Find/estimate variance

Let $w_{11},\ldots , w_{nm}\in [0, 1]$ be a set of constants and $H_1(t), \ldots , H_m(t)$ be some cumulative distribution functions (CDFs). Consider a sample of independent random variables $\xi _1, ...
0
votes
3answers
35 views

You roll 3 six-sided dice. What is the probability that the third is at least as high as the highest of the previous two?

I know that the probability of the first two dice being different is $\frac56$, and the first/second being greater is $\frac56$, but am not sure how to calculate the prob of the 3rd being greatest. ...
0
votes
1answer
19 views

Cumulative Distribution Funciton to pmf

I am still quite new to cdf and pmf. When we only have pmf for x = 1, 2 and 4 , how should I understand the corresponding cdf as in the pmf for x = 3 doesn't exist. Also I tried to draw the piecewise ...
0
votes
0answers
20 views

Calculating expectation of random vector

Let $\Omega=\Theta\times \Pi$ be a finite sample space and $P$ be prob. measure on $2^\Omega$. We define random vector $X: \Omega \rightarrow \mathbb{R}^n.$ How can I calculate following conditional ...
-3
votes
1answer
32 views

Unfair coin tossed twice

An unfair coin is tossed twice. The probability of heads is 3 times the probability of tails. What is the probability that at least one head is flipped?
0
votes
1answer
23 views

independence of random variable

Suppose we have $2$ Independent random variables $X$ AND $Y$. Let $f(X)$ and $g(Y)$ are functions of those $2$ random variables. 1.) my question can we say that the functions $g(X)$ AND $f(Y)$ are ...
0
votes
0answers
27 views

Notation in probability theory: conditional on multiple events or joint of event with an conditional one

It might be a quite dumb question and if so, I apologize in advance (I am kind of a newbie in probability theory ). But once in a while it bothers me and I can't find the answer to it. Ok, now the ...
1
vote
0answers
19 views

Conditioning on Brownian motion

I was reading on conditional probability with respect to a partition of a sample space, and I came across the following example: Let $(N_t:t\geq0)$ be the Poisson process. Given fixed times $0\leq ...
0
votes
0answers
18 views

Markov Chain with dependence between users

I am looking for a Markov Chain model that describes the following problem. I have $N$ indifferent users in the system, each of them has three states: $A$, $B$, $C$, and I know the transition ...
2
votes
1answer
29 views

Expected value of room enters

I was looking at previous exam questions, but one of the questions I don't know how to solve correctly. In this question I need to calculate the expected amount of rooms the mouse enters before he is ...
0
votes
0answers
17 views

Bayesian Updating - plug in previous posterior for prior?

Let's say I have two sequences of observations, $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$. For each sequence I'm going to estimate the probabilities of certain events occurring, namely event $A$ in ...
1
vote
1answer
22 views

Infinitesimal Generator of Poisson process

I would like to compute the infinitesimal generator of a Poisson process $N$ with intensity $\lambda$. So I can write: $$\mathbb{E}[\ f(N_{t+s})-f(N_s)\ |\ \mathcal{F_t^0} \ ] = \mathbb{E}[\ ...
0
votes
0answers
26 views

Number of permutations on nearest neighbors

Consider a finite connected set $A \subset \mathbb{Z}^d$ and let $S_A$ be the set of permutations on nearest neighbors. Namely, the elements of $S_A$ are bijections $\pi : \, A \rightarrow A$ such ...
0
votes
3answers
39 views

Can you simplify this expression?

This is a Bayes formula incorporating 2 random variables. The final expression seems a bit tricky to simplify the exponents and I'm still not so confident with my algebra (pardon me ;)). Can you have ...
0
votes
2answers
20 views

Probability of two strings being equal

Given a matrix $A\in F_2^{n\times m}$, (let $m< n$ and $A$ has full column rank) what is the probability under the distribution ( $y,y'$ uniformly random in $\{0,1\}^m$), such that $Ay=y'$? I am ...
0
votes
2answers
31 views

Tabulate the probability distribution of $x$.

If a red dice and a green dice are rolled together and $X$ is the highest score minus the lowest score of the dice, what are the possible values of $X$? Tabulate the probability distribution ...
1
vote
3answers
62 views

Sample space: What's the possibility that a family has n boys?

What's the sample space in these two cases? Case1: Of all families with two children you ask the parents, if they have a boy born on a Thursday. They say yes. What's the possibility that the family ...
0
votes
1answer
14 views

What's the probability, and how to choose the right formula?

Question 1: Toss a coin 4 times. Let $A$ denote the event that a head is obtained on the first toss, and let $B$ denote the event that a head is obtained on the fourth toss. Is $A \cap B$ empty? ...
0
votes
1answer
13 views

How to prove that expectation is the integral of survival function? [duplicate]

I am trying to prove that $E[X] = \int_0^{\infty} P (X > x)$ I have started like below: $$\text{E}[X] = \int_{0}^{\infty}x f_{X}(x) dx $$ $$ = \int_{0}^{\infty}\int_0^x dy f_{X}(x) dx $$ ...
2
votes
3answers
50 views

If a die is thrown thrice. Find the probability that the largest score is three times the smallest.

I have no idea about the answer, but I'm viewing the question this way; If the smallest score obtained from the any three throws of the die is $1$, then largest among the other two throws must be ...
1
vote
1answer
26 views

How to calculate the probability of a random variable given two independent variables?

For example, to calculate $\mathbb{P}(C \mid X_1,X_2)$, I know $\mathbb{P}(C \mid X_1)$ and $\mathbb{P}(C \mid X_2)$. $X_1$ and $X_2$ are independent random variables. If there is a way to calculate ...
0
votes
1answer
13 views

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, ...
6
votes
5answers
360 views

Is there an alternative intuition for solving the probability of having one ace card in every bridge player's hand?

I am trying to get to know probability a little better since it's a weak point for me and I was wondering what other ways there were to intuitively think about the problem of finding the probability ...
2
votes
4answers
24 views

conditional probability of several events

I'm having a hard time understanding what this question wants: A person initially purchases either type A or type B. She will choose either type A or type B with an equal probability on her first ...
0
votes
2answers
36 views

Calculating the Variance of a Dice Roll?

Here's my thinking: $$Var(X) = E(X^2) - E(X)^2$$ Assuming each roll is independent: $$E(X^2) = E(XX) = E(X) \cdot E(X) = E(X)^2$$ Thus: $$Var(X) = 0$$ However, this is not correct. Where did I ...
1
vote
1answer
26 views

Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = 1$, for $0 < xi < 1$ for $i = 1, 2$.

Find: (a) $E(X_1 X_2)$, and (b) $Var(X_1 X_2)$. Isn't (a) = zero, since this are independent? How do I go about (b)
1
vote
1answer
21 views

Best algorithm for finding permutations with constraint of average total value.

Let's assume I have a random number generator from 0-100 included (only integers) and I generate 5 numbers with it. I want to know the probability of hitting 80, 80, 80, 80, 80 with the constraint ...
1
vote
1answer
19 views

Optimize order of a list based on time to complete, probability of success

I'm a programmer participating in a coding challenge, but I'm not up on my advanced math. I'm currently working on a solution to a problem, and have a semi-functional program, but I'm still missing a ...
1
vote
1answer
17 views

Notation and a problem with Aleatory variables (Advanced Probability)

I am studying advanced probability and I have a question with notation. One exercise says: Let $(\Omega,B)$, show that $A \in B$ iff $1_A \in B$. But, $1_A$ is a function, what the book means with ...
4
votes
1answer
37 views

an exercise related central limit theorem

I'm working on the following problem in Durrett: Let $X_1, X_2, ...$ be i.i.d, nonnegative, $EX_i=1$ and $Var(X_i)=\sigma ^2$. Then we have $2(\sqrt{S_n}-\sqrt{n})$ converge to $\sigma \chi$ in ...
3
votes
1answer
32 views

Limit (in probability) of sequence of independent random variables

We have $\{X_n\}$ independent random variables which converge to $X$ in probability. I was asked to prove that $X$ is constant. My approach is to try to show that$Var(X)=0 \implies X$ constant, but i ...
3
votes
2answers
46 views

Markov Chains - Strong Markov Property

I'm struck with an exercise. I tried, but the results don't seem to fit to those proposed. Exercise: Two players play the following game. The one who begins draws two cards from a deck of 40 cards ...
1
vote
1answer
34 views

Let X and Y be a random variables with $E(X) = 5$, $Var(X) = 30$, $E(Y ) = -􀀀5$, $Var(Y ) = 10$ and $Cov(X, Y ) = 7$

(a) Find $E(2X-3Y+1)$. (b) Find $E((X-2Y)^2)$. (c) Find $Var(3X-Y+pi)$ First I found $E(X^2)$ and $E(Y^2)$ using the given values for (a) I have $2E(X)-3E(Y)+1$ for (b) I come up with: ...
1
vote
0answers
11 views

Doob's submartingale stopping theorem in the context of the submartingale problem

Let $$X^\omega_f (t, w) = f(w(t)) - f(w(t \wedge \tau)) - \frac{1}{2} \int_{t \wedge \tau}^t \Delta f(w(s))\, ds$$ be a $P^\tau_\omega$-submartingale. 1) Why Doob's submartingale stopping theorem ...
0
votes
1answer
22 views

Law of Iterated Expectation with Probability?

I'm trying to follow a proof of the following proposition (source) Let X and Y be two independent random variables and denote by $F_X(x)$ and $F_Y(y)$ their distribution functions. Let $$Z=X+Y$$ ...
1
vote
3answers
67 views

probability and expected value

Hey I am not sure if I thinking correctly on this question? In a carnival, there is game which charges you $3$ dollars to play a game. You win $1$ dollar for every consecutive head you get and you you ...
1
vote
2answers
44 views

Showing that infinite product of random variables goes to zero: $\prod^\infty X_i \rightarrow 0 \text{ a.s.}$

I am doing the following exercise: Let $X$ be a strictly positive rv with $\mathbb E[X]=1$ but $X \neq 1$ almost surely. Let $X_1, X_2 \dots$ be iid with same distribution as $X$. Now let $M_0=1$ and ...
3
votes
3answers
70 views

How biased is this biased coin

Suppose that we have a coin that we suspect is biased, but that we don't know precisely how biased it is: all we know is that its probability p of landing heads is some fixed value between .4 and .6, ...
0
votes
0answers
23 views

How to solve following sequence equation?

Suppose I have a sequence ${p^*_j}$, $j=0,...,m$, satisfying relations $p^*_j=p^*_{j-1}p_{j-1,j}+p^*_{j}p_{jj}+p^*_{j+1}p_{j+1,j}$, with \begin{equation} p_{jj}=\frac{2j(m-j)}{m^2}, \end{equation} ...
1
vote
1answer
24 views

CDF and Convergence of Maximum of Sequence of i.i.d. R.V. of Random Length

Let $X_1,X_2,...$ be i.i.d random variable $U(0,1)$ distributed. Let $N_m$ be $Poisson(m)$ and independent of each $X_i$. i)Find the cumulative density function of ...
1
vote
3answers
33 views

probability of exactly one out of N events occuring

I have N events. Each "i" event has probability $P_i$. What is the probability of $n$ events occuring? I have seen this answered for two and three events, but not for an arbitrary N. In principle, ...
1
vote
1answer
34 views

Roulette with p=$\frac{2}{3}$. What is the probability of not going home?

I'm learning about the gamblers ruin. The problem is that I don't know how to calclate the formula. I got two exercise questions in my book. Both of the questions will be about a strange roulette ...