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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Normal distribution

can anyone help me calculate $E(Z^4)$, $E(Z^3)$ for $Z\sim N(0,1)$? I know that $Z^2\sim \chi^2(1)$ then $E(Z^2)=1$, $Var(Z^2)=2$. Thank you.
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0answers
9 views

What is probability that a team reaches final if we know the probabilities of all opponents in the semi-final?

Our Discrete Math professor asked us a question as the Euros are going on. Given the following info, what is the probability that Portugal will make it to the final? Win Probabilities in quarter ...
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2answers
64 views

How to understand this integral result?

I was reading this page on Wikipedia: Birthday Attack I can understand up until how to approximate the minimal number of attempts for a given probability $$n(p; H) \approx \sqrt{2H \log \frac 1{1-p}...
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Ito's Formula applied to a weird equation…

I was just wondering if someone could explain how to solve this problem. I have an equation $X(t, S_{t}) = s\partial_{s}u - u$ where we have $u(t,S_{t})$. Now, according to the paper, evaluating $dX$ ...
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0answers
16 views

Martingale system expected winnings

Suppose we have a starting amount of money and gamble them using the martingale system by doubling the starting bet on a loss and resetting to out starting bet on a win. How can we calculate the ...
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1answer
52 views

Modified gambler's ruin problem: quit when going bankruptcy or losing $k$ dollars in all

In each round, the gambler either wins and earns 1 dollar, or loses 1 dollar. The winning probability in each round is $p<1/2$. The gambler initially has $a$ dollars. He quits the game when he has ...
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1answer
173 views

Is Lottery probability really the same for all combos?

http://justwebware.com/uklotto/uklotto.html Test run quickpick Test run 1,2,3,4,5,6 Test run (single digit,teens,twenties,twenties,thirties,forties) 1000 times or more each cycle for as many ...
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3answers
28 views

Expectation over sequencial random shuffles

I'm trying to understand this concept with this following problem: Logan is cleaning his apartment. In particular, he must sort his old favorite sequence, , of positive integers in ...
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1answer
2k views

Probability: Drawing Aces from a Deck of Cards

"You are dealt 13 cards randomly from a pack of 52. What is the probability your hand contains exactly 2 aces?" I thought about breaking it down into: ${4 \choose 2}$ = number of ways to choose two ...
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2answers
27 views

Find $a$ so that $a(e^{-2x}-e^{-3x})$ is a probability density function. [on hold]

Let $f(x) = a(e^{-2x}-e^{-3x}),$ for $x\geq 0$, and $f(x) = 0$ elsewhere. (a) Find $a$ so that $f(x)$ is a probability density function. (b) What is $P(X\leq 1)$? Image. If it is possible, ...
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1answer
21 views

Expected value for a sorted collection

We have a collection of n numbers (duplicates can appear). We try to sort the collection by randomly shuffling the collection. What is the expected number of times we have to shuffle the collection to ...
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2answers
29 views

Which probability is greater, given minimal info

which probability is greater, given that $X$ and $Y$ are independant, positive random variables? There is also the option that it's impossible to know as we don't have enough information. I'd ...
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1answer
32 views

Is it possible to solve for a variable without specifying a probability distribution?

Is it possible to arrange this simple equation into a form that will allow me to not specify the distribution. I am not quite adept with characteristic functions so I am not sure if they help in this ...
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2answers
49 views

Why is the probability density function of a normal distribution exponential? [on hold]

I came across this while self-studying for a probability course but I still don't quite get the rational behind it. Would appreciate it if someone could provide some intuitive explanation or rigorous ...
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3answers
35 views

Concerning The Number of Ways of Drawing a Full House vs. Two Pair

The Wikipedia entry for "Poker probability" gives the following result for the number of ways of drawing a full house: $$ \binom{13}{1} \binom{4}{3} \binom{12}{1} \binom{4}{2} = 3, 744. $$ The logic ...
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0answers
8 views

HMM Training: Testing convergence

What is the best way to test for convergence while training an HMM? I understand that we need to iterate till the change in parameters ( transition matrix, emission matrix ) is less than the threshold....
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0answers
29 views

What is the expected number of coin flips when using a branching algorithm?

Suppose I am using this algorithm: Given an unbiased coin, flip it. If it is heads, apply this algorithm to two unbiased coins. Supposing also that I begin by applying this algorithm to a ...
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1answer
29 views

Throwing dice and finding limits

We throw an fair dice $n$ times. Let $S_n$ be the number of throws with even number of dots on the dice. 1.) Calculate the limit $$\lim_{n\rightarrow\infty}P(2S_n \leq n)$$ 2.) Express the value of ...
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0answers
34 views

Kolmogoroff 0-1 does this proof work?

I have thought at this proof of the Kolmogorov 0-1 Law varying a little the sketch found in Probability essentials (Jean Jacod, Philip Protter). My questions are Is it a valid proof? Is it a bad ...
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3answers
33 views

given the following CDF, find the expected value

I got stuck at the middle of the question. would appreciate your help. first of all, given the CDF as follows, I had to find parameters $a$ and $b$ such that the CDF is a function of a continuous ...
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0answers
23 views

Probability of Elements belonging together

Imagine a camera scene, where an algorithm labels the person which are inside from 1 to n. Now, imagine there is not just one perspective, but multiple. That means multiple cameras looking at the same ...
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0answers
12 views

Help with Boolean algebra

Consider a system with $n$ units where each unit is either working or failing. $x_j=1$ represents the condition that $j$-th unit is working. Suppose each unit is working with independent probability $...
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1answer
57 views

Monotone Class Theorem and another similar theorem.

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\...
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1answer
39 views

Determine probability based on observation

Suppose there is an urn with 100 balls, of two colors, say white and black. Let $p$ be the probability of drawing a white ball. You draw one ball, replacing after the draw. After 100 draws, each with ...
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1answer
663 views

Conditional expectation of the sum of three dice rolls given the sum of their maximum and product

Consider the random experiment in which three fair dice are rolled simultaneously (and independently). Let $X$ be the random variable defined as the sum of the values of these three dice. Let $Y_1$ ...
4
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1answer
123 views

Weak Law of Large Numbers

The Weak Law of Large Numbers is often stated with the iid assumption for the underlying RV's. However, I have seen the independence assumption being diluted to the "uncorrelatedness" assumption (e.g.,...
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0answers
33 views

Expected number of balls from throwing between boxes

I have 6 boxes: $A,B,A',B',C \text{ and } D$. The box $A$ has $n_1$ red balls that are numbered from $1, \cdots, n_1$. The box $B$ has $n_2$ green balls that are numbered from $1, \cdots, n_2$. Make a ...
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1answer
24 views

Conditional Probability for a Poisson Distribution: X = 1 | X $\geq$ 1

Suppose X has a Poisson distribution with a standard deviation of 4. What is the conditional probability that X is exactly 1 given that X $\geq$ 1? I know that for this problem $\lambda$ is 16 ...
2
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1answer
31 views

Probability of one Poisson variable being greater than another

Given two Poisson distributions with different λ values, if each were to produce a single random variable, is there closed-form expression for calculating the probability of one random variable being ...
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0answers
23 views

Compute conditional probability

If a conditional probability table is given for $P(S_t|M,E)$. How to compute the value for $P(S_t = x | M,E)$ ? where $E$ is binary (0 or 1) and $M$ is ternary ?
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0answers
15 views

Green's function and strong Markov property for stopped Brownian motion

Let $X(t)$ be a Brownian motion in $\mathbb{R}^n$, stopped at some fixed time $T$. Is there a notion of Green's function for such a Brownian motion? I am guessing that there is, and $G(x, y) : = \...
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4answers
94 views

why is Probability (at least one solved$ =$P(A\cup B)$

I have a question in which it is stated that the probability of a student solving a problem A is $\frac{2}{3}$. And the probability of solving another problem B is $\frac{3}{5}$. So what is the ...
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0answers
8 views

How much larger is the Likelihood Function Under True Model?

Let $X$ be a random variable with probability distribution functions given by $f$, and let $g \neq f$ (on a set of positive measure) be some other distribution. $D=\{x_1, \ldots, x_n\}$ is a set of $n$...
3
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1answer
78 views

Expected number of steps - shuffling a sequence

I've been struggling with a problem a CS student friend of mine gave me a few hours ago. Given that $P$ is an array of integers and $N$ is its size, how many minutes is the following algorithm ...
2
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1answer
53 views

$X_n$ Poisson independent, $\mathbb{E}[X_n] = \lambda_n$. If $\sum \lambda_n = +\infty$, then $\frac{S_n}{\mathbb{E}[S_n]} \rightarrow 1$ a.s

Let $X_n$ be independent Poisson random variables with $\mathbb{E}[X_n] = \lambda_n$. Define $S_n = X_1 + \dots + X_n$. Show that if $\sum \lambda_n = +\infty$, then $\frac{S_n}{\mathbb{E}[S_n]} \...
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1answer
28 views

Expected number of permutations required to sort a list of numbers

Given a list of N numbers if we are performing random permutations each time and checking whether the list is sorted, what will be the expected number of permutations required to sort that list?
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1answer
518 views

Different Perspectives of Multinomial Theorem & Partitions

There are 2 important interpretations of the multinomial theorem and coefficients. 1: Determining the number of ordered strings that can be formed using a set of letters. For example, with 1 m, 4 i'...
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4answers
26 views

Expected value of a roll of a fair die given that the number rolled is at least 4

I am trying to understand the solution to a probability problem, and I am having trouble understanding where some of the numbers are coming from. The textbook gives this definition for conditional ...
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0answers
30 views

The probability for a team to win the cup in a championship. A simplified case.

Hypotesys: For each two national football teams $i$ and $j$ that play against each other the probability of victory for $i$ is defined as: ${{p}_{i}}=\frac{{{N}_{i}}}{{{N}_{i}}+{{N}_{j}}}$, where $N$...
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1answer
1k views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
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0answers
28 views

Birthday Problem Variant: Probability of exact number of people sharing a birthday

I'm working with some data that includes a person's date of birth. The list includes 2500+ unique individuals and using Excel it's very easy to count the number of people who share a birthday with at ...
6
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2answers
97 views

probability of sorted array with duplicate numbers

Suppose I have a sequence of n numbers {a1,a2,a3,...an} where some of the numbers are repeated. What is the probability that the sequence is sorted?
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2answers
23 views

Brownian Motion-Independence of Increments

Consider a Brownian Motion $B(t)$ with $B(0)=0$. Suppose $s<t$. I read in a book that while $B(t)-B(s)$ is independent of the past, $2B(t)-B(s)$ or $B(t) - 2B(s)$ is not. Why is this the case? ...