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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Cashier has no change… catalan numbers.. probability question

I think this question uses catalan numbers.. but I don't know exactly how to answer it... its not homework or anything but I need to understand how to do it.. I tried drawing up likes for each 5r ...
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2answers
18 views

Expected value with two uniformly distributed random variables

A surveyor wishes to lay out a square region with each side having length L. However, because of a measurement error, he instead lays out a rectangle in which the north–south sides both have length X ...
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0answers
2 views

Simple lower bound for the connective constant of the plane square lattice for self avoiding walks

We know that the connective constant of plane SAW (Self Avoiding Walks) on the square lattice is between 2 and 3. There are very accurate estimations of this constant. It's very easy to see that it ...
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0answers
5 views

proof of a special case of discrete-time tower property

I'm reading a book on stochastic process and the first chapter is about properties of conditional expectation. One of the example the book gives is the proof of a special case of tower property in ...
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1answer
22 views

Proof about independent random variables

Let $X_1,X_2,...$ be independent random variables with $P(X_n=1)=p_n$ and $P(X_n=0)=1-p_n$ Show that $X_n\rightarrow 0$ in probability if and only if $p_n\rightarrow0$, $X_n\rightarrow 0$ almost ...
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4answers
25 views

Probability Question: Dividing Groups [on hold]

There are 4 boys and 12 girls in a class. They are randomly divided into 4 groups of 4. What is the probability that each group contains a boy?
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0answers
6 views

Probability Generating Functions- Dependent Poisson Distributions

I was wondering if anyone could give me a tip on how to proceed with the following question? Suppose X~Poisson(N), where N~Poisson($\lambda$). What is the PGF of X + N? (Where $\lambda$ is a number) ...
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2answers
38 views

Find the pdf of $C(X)$

Suppose that the probability of $x=0$ is $p$, and the probability of $x=1$ is $1-p=q$. Consider the random sequence $X=\{X_i\}_{i=1}^{\infty}$. We map this sequence by $C$ to a point in the interval ...
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0answers
6 views

Sufficiency of xbar in a binomial(2,p) population.

Let $\{X_1,X_2,...,X_n\}$ be a random sample from a $\text{Bin}(2,p)$ population. Use the definition of sufficiency to show that the sample mean is sufficient for $p$. Here I am not allowed to use ...
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0answers
16 views

Finding the density function of a function of random variable $X$

For two distinct density functions $f_0$ and $f_1$, log likelihood ratio is given as $$l(x)=\log\left(\frac{f_1(x)}{f_0(x)}\right)$$ Let $f_0\sim\mathcal{N}(-0.9,1.1)$ and ...
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0answers
24 views

Probability in Combination of 5 colour in 9 spaces

We have 5 colours: red, green, blue, black and white, and 9 spaces to paint with only one of that colours each. What is the probability of having 5 spaces in white and the other 4 all in different ...
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1answer
25 views

Find $ 1 - a^x$, where $x$ is a random variable

I'm trying to find the value of $$ 1 - \left(\frac{99}{100}\right)^N, $$ where N is a random variable given by $$ P(N = i) = \frac{e^{-(i-\lambda)^2 / 2\lambda}}{\sqrt{2\pi\lambda}}, $$ with $0\leq ...
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1answer
33 views

Back to square 1…

A friend of mine was telling me about one of the problems, which he described thus: As you can see, the answer to the toy problem presented here is reportedly 13. However, I don't understand how ...
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2answers
18 views

Posterior probability or Prior probability

I have an arguement with my friends on a probability question. Question: There are lots of stone balls in a big barrel A, where 60% are black and 40% are white, black and white ones are identical, ...
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0answers
9 views

How we do that to determine the independent variables in data?

I have a data about PM10 (particul matter in air). Now PM10 is my dependent variable and i have to determine that which variables are the best independet variables in my data-set for PM10? I using R. ...
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1answer
21 views

Show that Kolmogorav's axiom implies that $P(A) \le 1$ for any $ A$

Show that Kolmogorav's axiom $P(A) ≥ 0, P(A ∪ B) = P(A) + P(B)$ if $A\cap B = ∅$, $P(S = 1)$ implie that $P(A) ≤ 1$ for any $A$.
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5answers
33 views

Throw a pair of dice 60 times. What is the probability that the sum 7 occurs between 5 and 15 times?

Throw a pair of dice 60 times. What is the probability that the sum 7 occurs between 5 and 15 times? I know this is how you could calculate the probability of sum 7 occurring 5 times: (60 choose ...
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1answer
16 views

Bounding $\mathbb{E}(X_{i_1}\cdot \ldots \cdot X_{i_k}) $

Consider random variables $X_1,\ldots X_n$ with zero mean, variance at most $1$, $k$-wise independent $k\leq n $ and bounded: $|X_i|\leq C$ for some $C\geq 1$. If I assume $k$ is even, how can I ...
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2answers
19 views

Proving $P(A|(B \cap C)) = P(B | (A \cap C)) P(A | C) / P(B | C)$ using Bayes' theorem.

The following equation can be proven rather uglily, provided that $P(B \cap C)$, $P(A \cap C)$ and $P(C)$ are non-zero, by expanding the conditional probabilities. $$P(A | (B \cap C)) = \frac{P(B | ...
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0answers
20 views

Expectation of product of cosines

I am reading a paper that starts with $$ E[ \cos( a(x-y) ] = E[ \cos(a x) \cos(a y) + \sin(a x) \sin(a y) ] $$ where the expectation is over $a$, then converts it into something of the form $$ = 2 ...
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2answers
25 views

Show $E\left(\mathbf{X}_i \otimes \mathbf{u}_i\right)=\mathbf{0}$ implies $E\left(\mathbf{X}_i^{\top}\mathbf{G}\mathbf{u}_i\right)=\mathbf{0}$

Let $\mathbf{X}_i$ be a $G \times K$ random matrix, and let $\mathbf{u}_i$ be a $G \times 1$ random vector, and suppose we have a sample of $i=1,\ldots,N$ of each. Suppose the following condition ...
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0answers
15 views

pROBABILITY USING DICES [on hold]

When dice are irregular so that the sides of the dice are not equal in size or weight, then the most accurate way to determine the probability that they will land with a certain side (such as 5) up is ...
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0answers
5 views

inter-event time distribution

We have a counting process N(t) and two processes X(t) and Y(t) where each renewal point of N(t) is a renewal point of X(t) with probability q or Y(t) with probability 1-q. If Fn(t) is the ...
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0answers
11 views

Transition probability of Brownian Bridge

This is homework so no answers please Consider Gaussian $X_i\sim N(0,t_{i}(1-t_{i}))$ s.t. $\frac{X_{1}}{(1-t_{1})}$ and $\frac{X_{2}}{(1-t_{2})}-\frac{X_{1}}{(1-t_{1})}$ are independent (with ...
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0answers
24 views

Probability of upper quartile student

A teacher was asked by her principal to select 7 students at random from her class to take a standardized math test.The teacher previously had rank ordered her students on the basis of their ...
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1answer
19 views

Possible/Impossible Probability Question

I'm not sure if there is a question like this already here, but... I just thought of a question related to probability, and I was wondering if it was possible: Suppose you want to ask someone to ...
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1answer
26 views

probability of teacher selecting students [on hold]

A teacher was asked by her principal to select 6 students at random from her class to help out on an outing to a senior's home. In her class, she has 6 girls and 4 boys. The principal believes that ...
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1answer
17 views

Expectation under two equivalent probability measure

For two probabilities, if they are equivalent, then does there exist a r. v such that the signs of its expectation under those two probabilities are different?
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1answer
29 views

A gambler who is equally likely to either win or lose one unit

A gambler who is equally likely to either win or lose one unit on each gamble will be down n before being up 1 with probability 1/(n + 1); or equivalently, P(gambler is up 1 before being down n) = ...
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0answers
19 views

Estimate on the Positive probability of not hitting finite measure sets in $\mathbb{R}^{d}$

In $d\geq 3$, we have that BM is transient a.s. i.e. $\lim_{t\to \infty}|B_t|=\infty$. But does this imply $1-P_x(T_A<\infty)>0$ for Borel sets $A\subset \mathbb{R}^d$ with ...
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6answers
46 views

Expectation of two dice game

The game plays like this: You roll two dices at the same time. If you get same number on both dices, you have to roll again, until you get different numbers. If you get different numbers, you stop. ...
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2answers
31 views

In Texas Hold'em poker, is the ranking according to chance of beating 1 opponent's hand the same as according to beating multiple opponents?

In Texas Hold'em poker, you can rank hands according to the probability of beating one randomly generated opponent's hand. However, no one can compute the exact probabilities of beating $8$ random ...
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1answer
8 views

Bernoulli trials case in probability

A fair die is tossed twice. About how many times would you expect to roll 3 or greater? So based on sequence of Bernoulli trials: P(exactly k successes in n trials) = C(n,k) p^k q^(n-k) where p = ...
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0answers
26 views

probability distribution function of two independent variables

Let $X$ be a random variable whose distribution function is $F_X(t)=3^{-t}$. Suppose that $Y$ is another random variable whose distribution function is $F_Y(t)=4^{-t}$. What is the probability that at ...
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4answers
37 views

Expected number of rolls on a dice?

You roll a die until you have seen a 5 on 4 of the rolls (e.g. ⟨5,3,2,5,4,1,6,5,2,5⟩. What is the expected number of rolls this will take? I think that I am way overthinking how I should be going ...
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1answer
12 views

Covariance matrix of Y when we have the covariance matrix of X

If the random vector $\mathbf{X}$ is transformed according to \begin{align*} Y_1 &= X_1\\ Y_2 &= X_1 + X_2 \end{align*} and has a covariance matrix $$ \mathbf{C}_X = ...
0
votes
1answer
17 views

Solving inhomogenous first order difference equation (recurrence relation)

I have the equation (arising in a probabilistic context) $$ x_n = a(1-x_{n-1}) + (1-a)x_{n-1} $$ and I'm told that there is a solution of the form $c_1 + c_{2}\lambda^n$. How do I solve it, i.e. how ...
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0answers
19 views

Likelihood interval

Assume that $Y_1, Y_2, \ldots , Y_n$ are independent Poisson random variables each with rate parameter $\lambda$. Assume $n$ is sufficiently large enough so that the central limit theorem (CLT) ...
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1answer
28 views

Statistics: Simple pick from bag problem

I am doing a personal project with neural networks and want to see how accurate the predictions are compared to just plain old guessing. I'm sure this is a typical probability textbook problem, but I ...
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0answers
6 views

How to generate a random vector with a triangular autocorrelation

I am trying to generate N random vectors that when correlated with themselves converge to a triangular function. The approaches I have been trying to follow are reversing a PCA procedure ...
2
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0answers
17 views

Approximation of minimum among many binomials

We choose $k$ numbers independently from the binomial distribution $B(n,1/2)$, where we can think of $n$ as large. What is the expectation of the minimum of the $k$ numbers? Is there a good way to ...
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0answers
14 views

Is this application of the law of total probability correct?

Let us consider a counting process $N(t)_{t\geq0}$ which is neither Markovian nor Levy. Is it correct to write $$ \mathbb{P}(N(t)=j)=\int_{0}^{t}\mathbb{P}(N(t)=j, N(s)=i)ds $$ for $j\geq 1$ and ...
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0answers
13 views

What is the definiton for “best probability measure”?

I'm looking for this definition is notes that use the phrase and elsewhere, but it just isn't there. Does anyone else recognize the phrase?
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2answers
30 views

Finding the total permutations of the cards in the hand.

There are $36$ unique cards containing $9$ ranks ($1$-$9$) of $4$ suits (diamonds, hearts, clubs, and spades). A hand is a collection of $9$ cards. The hand must contain all $4$ of the $1$s (one from ...
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1answer
42 views

The probability that $3$ random points on the circumference form a right-angled triangle?

In my probability theory course, I dealt with a similar problem which asks for the probability that $3$ random points on the circumference of a circle lie on the same semi-circle. But it makes me ...
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1answer
34 views

Help with Probability

Hey Im a new guy here and need some help. I have an assignment bugging me. I can't really figure out which why to go around it. I'm thinking conditional probaility but how to apply the Bayes Theorem ...
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0answers
40 views

Checking my solution for a probability question? [on hold]

Hi, I have unfortunately lost my solutions. I got (i) which is 9!, but there are no answers for the second question. I stated that P(none together)=1-P(3 together)-P(2 together) and got the ...
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2answers
63 views

Number of ways to seat people around a circular table

I got (i) which is $9!$, but there are no answers for the second question. I stated that $$P(\text{none together})=1-P(\text{3 together})-P(\text{2 together})$$ and got the answer $7/12$. Is this ...
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1answer
21 views

what do these odds ratios represent?

I am reading this article in which is given the matrix of the joint probabilities of two random variables, X=$(x_1,x_2)$ and Y=$(y_1,y_2)$. Let's say they are $(p_{1,1},p_{1,2},p_{2,1},p_{2,2})$. ...
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1answer
19 views

is it true that conditional expectation Y to X is a function of X?

I mean, is it true that $E(Y|X) = \phi(X)?$ if so, how should we derive the form of X?