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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1answer
7 views

Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
2
votes
1answer
16 views

Proof of equivalent probabilities in anomaly detection

In A New Look at Anomaly Detection there is a claim for the proof of probabilistic definition of normal is as follows, a guess of the probability for event i is $\pi_i$, the true probability is ...
0
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1answer
14 views

sum of discrete and absolutely continuous random variables

We know a sum of two independent absolutely continuous random variables is still absolutely continuous since its density function is the convolution. And without being independent, the sum may be ...
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0answers
27 views

Are all orthogonal projections conditional expectations?

When will orthogonal projections coincide with conditional expectations? Does that have something to do with the fact that not all closed subspace are probability spaces? Is it why when we fix a ...
0
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1answer
15 views

permutations vs combinations on slot machines with repeating elements on each reel

For a slot machine with 5 reels where there are repeated elements on each of the reel. Example: Reel 1 [ 1, 1, 2, 1, 3, 5, 6 ] Reel 2 [ 1, 2, 3, 4, 5, 5 ] Reel 3 [ 2, 2, 3, 2, 4 ] Reel 4 [ 1, 2, 3, ...
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1answer
43 views

Density of probability in a square

Suppose we have a square $$\{(x,y) : x \in [0,1], y \in [0,1] \}.$$ We suppose that we have $X$ and $Y$ are the coordinates in this square that are uniformly distributed. Why does the joint density is ...
1
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1answer
17 views

If $E(|X|)<\infty$, how do we show that it can be expressed as below

$F(x)$ is the distribution function of $\mathbb X$, and $f(x)$ is the derivation of $F(x)$, Prove that $\int_{0}^{\infty}(1-F(x))dx-\int_{-\infty}^{0}F(x)=E(X)$. Note that ...
1
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0answers
22 views

Probability with conditional involved

Alan is part of a dinghy sailing team. The probability of strong winds tomorrow is 0.3. In the strong winds the probability of Alan's team winning falls to 0.25. Calculate the probability that Alan's ...
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1answer
45 views

Probability a blackjack dealer will bust if you know their score and know the exact deck?

If you know the exact cards left in a deck, and the score of the dealer, how can you calculate the exact probability that they will bust? The dealer behaves as follows: If the dealer's score is less ...
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2answers
25 views

Prove that $X,Y$ are independent iff the characteristic function of $(X,Y)$ equals the product of the characteristic functions of $X$ and $Y$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $X$ and $Y$ be random variables on $(\Omega,\mathcal A,\operatorname P)$ with values in $\mathbb{R}^m$ and $\mathbb{R}^n$, ...
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3answers
70 views

How much would you pay to pull a ball from the bag?

If there are 9 white balls and one black ball in a bag. The white balls are valued at 10 dollars and black ball at 100 dollars. How much are you willing to pay for each pull from the bag (only one ...
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1answer
21 views

How to construct a transition matrix?

I'm giving my first steps in stochastic processes but I'm having some difficulties. See the following example Suppose that whether or not it rains today depends on previous weather conditions ...
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2answers
49 views

Why birthday distribution is not uniform. [on hold]

I was reading about birthday problem and I found a statement that real-life birthday distributions are not uniform since not all dates are equally likely (last line ...
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0answers
17 views

Probability of a decay process [on hold]

Consider a decay source with decay constant $\lambda$ (exponential decay) arriving with rate $R$ (poisson process). What is the probability that the product comes from the source right before it? (A ...
3
votes
3answers
154 views

How to understand independence of probability?

By definition, when $$P(E\,|\,F) = P(E)$$ holds, we say that $E$ is independent of $F$. By definition of conditional probability, $$P(E\,|\,F) = {P(E \cap F) \over P(F)} \Rightarrow P(E \cap F) = ...
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2answers
89 views

Game of probability

n a game, played between $2$ players there is a circular field and one of the players is blindfolded, who stands in the center of the field. The other player stands at a fixed point on the ...
0
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2answers
33 views

Confused by certain interpretation of expected value…

I read the following in Stein / Shakarchi's Fourier Analysis book, where they discussed the notion of expectation of a probility density. "Consider the simpler (idealized) situation where we are ...
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0answers
40 views

finding the limit for a martingale

I have trouble finding the exact limit for a martingale: Let $\{\xi_n\}_{n\in\mathbb{N}}$ be a Markov chain with $\xi_0$ uniformly distributed in $[0,1]$ and $$ ...
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votes
2answers
23 views

Calculate dependent probabilities

Imagine that two raffles will happen, every raffle will reward 1 person. 10 people will participate. The first raffle will reward 1 person. The second raffle will reward another person, but the ...
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0answers
20 views

Bayes theorem with multiple conditions

how to calculate P(a|b,c,d). knowing that b, c and d are NOT independent from each others ? i know how to solve it if there is independency assumption. however, i am just wondering if there is any ...
2
votes
1answer
22 views

Show that $|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$

Consider the random variables $X$ and $Y$ defined in the same space $\Omega$. Show that $$|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$$ This question comes from an old test, I know that ...
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votes
2answers
31 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
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1answer
20 views

The number of distinct values taken by a sequence of partial sums of iid

I'm working on an old exam problem as follows: Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of i.i.d taking values in $\mathbb{Z}$. Define $S_0 :=0$ and $S_n:=X_1+X_2+...+X_n,$ and $\theta_n$ to be ...
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1answer
28 views

Find the density

Suppose that radius $R$ of one sphere is a continuous random variable with density $$f_R(r)=6r(1-r) I_{[0,1]}(r)$$ Find $f_V(v)$ and $f_S(s)$ the densities of volume and surface area I did ...
2
votes
2answers
42 views

Probability of the simultaneous failure of two components where one component would take the full load if only one failed instead.

Working through an example question in Applied Probability for Engineers and Scientists, 1st Ed., by Ephraim Suhir. Example 1.9, beginning on p. 5, reads as follows: A heavy, nondeformable beam ...
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1answer
39 views

Expectation of a random variable in terms of its distribution function

Here is a theorem on expectation of a random variable in terms of its distribution function Theroem: Let $X$ be a (continuous or discrete) non-negative random variable with distribution function ...
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votes
2answers
34 views

A confusing probability question..

A magician holds one six-sided die in his left hand and two in his right. What is the probability the number on the dice in his left hand is greater than the sum of the dice in his right?
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3answers
55 views

Probability that an integer is divisible by $8$ [on hold]

If $n$ is an integer from $1$ to $96$ (inclusive), what is the probability that $n(n+1)(n+2)$ is divisible by 8?
0
votes
1answer
33 views

How do I determine the weight to assign to each bucket?

Someone will answer a series of questions and will mark each important (I), very important (V), or extremely important (E). I'll then match their answers with answers given by everyone else, compute ...
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votes
1answer
25 views

Poisson Distribution practice problem

I have just started to learn Poisson Distribution and I have no idea how to deal with the following practice from my textbook: Suppose the average amount of cars passing on a street per minute is ...
0
votes
1answer
30 views

Proof the statements

Proof the statements below i)If $P(A)=0$ and $B$ is any event, then $A$ and $B$ are independents ii)If $P(A)=1$ and $B$ is any event, then $A$ and $B$ are independents iii)The events ...
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votes
0answers
21 views

Probability of selecting a sequence in order

If "X" number of attempts are made by "Z" number of persons to select a random number from a range "r", where "X <= r". Then I am interested in finding the probability that a particular sequence in ...
1
vote
1answer
37 views

Does this game have infinite expected payout?

Consider the following game: Suppose the initial value of the pot is $ S $. Our player Josephine then rolls a fair $n$-sided die. If the roll is not $1$, then the pot is multiplied by that roll, and ...
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votes
1answer
35 views

Dice probability [on hold]

You roll a fair dice twice A. What is the probability that the first roll is odd and the second roll is even? B. What is the probability that one roll will be odd and the other roll will be even?
2
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2answers
51 views

Disjoint events

Let $A$ and $B$ two disjoint events such that $P(A)=0.3$ and $P(B)=0.5$. Find the probability that i)$A$ or $B$ ocurrs ii)$A$ occur but not $B$ iii)repeat $i)$ and $ii)$ with $A$ ...
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votes
2answers
17 views

Understanding different definitions of bayes theorem

I am taking course on probability and reading about bayes theorem. In Sheldon Ross' book, it given as $$P(E) = P(E|F)P(F) + P(E|F^C)P(F^C)$$ with note: Equation above states that the probability of ...
2
votes
1answer
28 views

Inequality for the derivative of a density of a random variable convolved with a normal r.v.

I have a question about the following proof. The statement is: Let $X$ be a random variable and $Z_\tau \sim N(0,\tau)$ be an independent random variable. Then $Y_\tau := X + Z_\tau$ has a ...
0
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1answer
29 views

Random variable with all higher order moments zero?

Is there a random variable with finite first and second moment but all higher order (non-central) moments are zero?
1
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1answer
64 views

Why does this expectation integrate to 1

Let $p(y|\theta )$ be our likelihood, and $\hat{p}_{N}(y|\theta)$ be an unbiased estimator of our likelihood. Let $z=\ln \hat{p}_{N}(y|\theta) - \ln p(y|\theta )$, and $g_{N}(z|\theta)$ be the ...
0
votes
1answer
15 views

Distribution of the summation of k random variables and k is also variable

We have a set of positive random variables $\boldsymbol X=\{X_1,X_2,\ldots\}$, where $X_1,X_2,\ldots,$ are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for Xi are ...
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votes
1answer
53 views

expected number of steps for chossing randomly each number between 1 to $n$ at least $k$ times [on hold]

Assume the following game: Every step choose a number between 1 to $n$ randomly i.e. every integer between 1 to $n$ is chosen with probability $\frac{1}{n}$. Success is when every number has been ...
2
votes
1answer
33 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
1
vote
1answer
33 views

Is this a binomial or multinomial question?

You can donate to a company: $10$ dollars , $20$ dollars or nothing. In a mall there are $70$% young people and $30$ % old people. $50$% from the old people aren't donating anything. ...
0
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2answers
32 views

Understanding Conditional Probability Basics

In many online sources I've read a statement similar to: Probability of B happening given A is equal to the probability of A and B both happening divided by B happening or $p(A | B) = p(A \cap ...
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1answer
45 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
3
votes
1answer
28 views

Probability of drawing >18 when drawing 3 cards

I am trying to calculate some probabilities for a card game. Players have to draw 3 cards each time and the cards must add up to a certain value for them to win - the value changes depending on the ...
2
votes
2answers
40 views

Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?

Here's what I have so far: All 4 same on first try = (1/4)^4 * 4 3 same, then get 4th on 2nd roll = 4 * (1/4)^3 * (3/4) * (4!/3!) Here's where I'm confused: 2 same = 4 * (1/4)^2 * (3/4)(2/4 :to ...
1
vote
1answer
31 views

Prove that $\tilde{X}_{\tilde{\theta}}(t)$ is a martingale

Let me introduce the objects: 0) $(\Omega, \mathcal{F},\Bbb{P})$ is a probability space 1)$S_N $ is the set of symmetric, non-negative definite $N\times N$ matrices 2)$a:[0, \infty) \times \Omega ...
0
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0answers
31 views

Probability of Probabilities :)

So here is a tough one (or so i think). i have 15 games (30 teams). and only 2 options i can chose from (even / odd number of goals). I want to bet a ticket with each possible combination. How many ...
3
votes
1answer
32 views

Conditional Probability for Exponential Random Variables

I'm working through a practice problem for an exam and I would like to verify that I've done it correctly. Additionally I'd like some insight on the intuition behind the numbers I'm getting. Problem ...