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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3
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1answer
51 views

Dense subsets of functional spaces

In books on Malliavin calculus and stochastic PDE, I found the following result is frequently used. I state it here in the simplest form. Given a separable Hilbert space $\left(H, \langle \cdot, ...
0
votes
0answers
28 views

If two poisson intervals overlap, does this affect the probabilities of $Y-X$?

Suppose I have two homogenous poisson processes with rate $\lambda$. $X = N(0,t_2]$ and $Y=N(t_1,t_3]$. I want to find the probability of $Y-X$. I know how to do this for when the intervals don't ...
1
vote
2answers
51 views

How can I get double expectation of two Random Variables?

Let random variables, $X$ and $Y$, be $X \sim \mbox{Geometry}(y)$ and $Y \sim \mbox{Uniform}(0, 1)$. How can I find out $E[X]$? The textbook said $$E[X]=\int_{0}^{1}\frac{1}{1-y}dy=\infty$$ Is ...
0
votes
1answer
23 views

Probability (Independent Events?)

I have no clue how to work out the two questions below, I would appreciate it if someone explained the method in detail. Question 6: Probability a) A couple move into a house. They wish to have a ...
-2
votes
1answer
42 views

Related to order statistics in probability theory

If $x$ and $y$ are two independently and identically distributed random variables then can we write the following? ...
1
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2answers
31 views

The relation between the discrete Harmonic function and the Harmonic function in PDE

Given a Markov chain with state space $\Omega$ and its transition matrix $P$, a function $h(x):\Omega\to\Bbb R$ is called a harmonic at state $x$ if $h(x)=\sum_{y\in\Omega}P(x,y)h(y)$, and is called ...
1
vote
1answer
36 views

The expected steps of two probabilistic events until a “success”

Problem: We want to eliminate $n\geq 0$ until $n \leq 0$ (Success). There are two things we can do with probabilities: $p_1$ Deduct 1 from $n$ $p_2$ Deduct $\frac{n}{4}$ from $n$ If $p_1 = p_2 = ...
8
votes
5answers
2k views

Draw cards repeatedly, until we find the ace of spades. Probability that we draw between 20 and 30 cards?

consider this problem: Draw cards repeatedly, without replacement, from a standard 52-card deck until we find the ace of spades. What is the probability that we draw between 20 and 30 cards? The ...
1
vote
2answers
40 views

Calculate $P(X > 1\mid Y = y)$ for both $x, y > 0$

The joint density is: $$f(x,y)=\frac{e^{-x/y}e^{-y}}{y}$$ Calculate $P(X > 1\mid Y = y)$ I'm going to consider the conditional density: I know that: $f(x\mid Y=y)=\frac ...
0
votes
0answers
19 views

averages of distributions easy

Suppose I go on the beach looking for treasure, and that the amount of treasure I find on a given day is given by a poisson r.v. with mean 50. Also suppose on average 4/5 of the treasure I find is ...
0
votes
2answers
79 views

Escaping Prisoner Probability Question

Question: A prisoner is trapped in a cell containing three doors. The first door leads to a tunnel which returns him to his cell after two days travel. The second door leads to a tunnel that returns ...
1
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2answers
31 views

Books on the same shelf

Three identical books were randomly tidied up in a cabinet containing five shelves, the probability that three books are on the same shelf is: My thoughts: I think the answer is $\displaystyle ...
0
votes
2answers
40 views

Different methods for the probability of choosing students for a committee

For this question, what I did was: $\frac2{18} \times \frac 1{17} \times \frac2{12} \times \frac1{11}$ The first two fractions are related to the choosing of girls and the next two are related to ...
0
votes
1answer
51 views

Variance of squared random variable

Can anyone help to prove this equation for any distribution $$ E(z^4)=1+\operatorname{Var}(z^2) $$ where $z$ is a random variable with the standard normal distribution $$z=\frac{x−μ}σ$$
0
votes
1answer
18 views

Regarding methods of enumeration in probability: Bridge (card game)

What is the probability of drawing a hand of 5 spades, 4 hearts, 2 clubs, and 2 diamonds in bridge? I believe the answer is: ...
2
votes
0answers
31 views

Finding the probability of a region $|X-Y|$

I have a region in a 3-D space with a density of $$ \ f_{x,y,z}(X,Y,Z) = \begin{cases} 1 & \text{if $ (x,y,z)\in W$}; \\ 0 & \text{if $(x,y,z)\notin W$};\\ \end{cases} \ $$ Being $W$ the set ...
1
vote
1answer
15 views

Finding the probability of a region inside a pyramid

I have a region in a 3-D space with a density of $$ \ f_{x,y,z}(X,Y,Z) = \begin{cases} 1 & \text{if $ (x,y,z)\in W$}; \\ 0 & \text{if $(x,y,z)\notin W$};\\ \end{cases} \ $$ Being $W$ the set ...
1
vote
1answer
28 views

Prove that there exists $s > 0$ such that $E[X^s] < 1$ given $E[X^r] < ∞$ for some $r$ and $E[\log X] < 0$

Let $X$ be a random variable with $X ≥ 0$ a.s. and such that $E[X^r] < ∞$ for some $r > 0$ and $E[\log X] < 0$. Prove that there exists $s > 0$ such that $E[X^s] < 1$. I know when $s$ ...
0
votes
0answers
57 views

How to identify what probability distribution this is?

in the context of a very long derivation (variational inference), by grouping the constant terms, I arrive at the following form that should be a probability distribution: $$ p(x) = C x^{-a} ...
3
votes
1answer
48 views

Expected value and a variance of a die sequence

There is a sequence of events that return a number. Here is a sequence: For a die with a probability of each side 1, 2, ..., 6 equal to ...
1
vote
0answers
51 views

Why does this generated $\sigma$-algebra contain all the sets that we have information about?

Assume that you have a collection of random variables $\{Y_\gamma: \gamma \in C\}$. Where each is a function $\Omega \rightarrow \mathbb{R}$. We define the $\sigma$-algebra: $\sigma(y_\gamma: ...
-1
votes
2answers
28 views

Expected Saving Value

I have 4 products and i have to choose exactly one product out of these four.So probability of choosing a product is 1/4. The value of money i save is ...
1
vote
1answer
102 views

How to interpret negative probability for a strategy in mixed nash equilibrium?

I am trying to get the mixed strategy in Nash equilibrium for the following matrix. $$\begin{pmatrix} 0 & 3 & 4 & 5 & 6 \\ 3 & 0 & 5 & 6 & 7 \\ 4 & 5 & 0 ...
0
votes
1answer
36 views

Probability Math Including Zero Probability Events

Consider two independent events, A and B. We would say the probability of A and B occurring is: P(A ∩ B) = P(A) * P(B) However, what if A is considered a zero ...
0
votes
1answer
49 views

Proof of Hoeffding's Covariance Identity

Let $X,Y$ be random variables such that $\text{Cov}\left(X,Y\right)$ is well defined, let $F\left(x,y\right)$ be the joint-CDF of $X,Y$ and let $F_{X}\left(x\right),F_{Y}\left(y\right)$ be ...
1
vote
1answer
17 views

Can a function $f:[0,2\pi] \rightarrow 1$ have a PDF and CDF?

Consider an interval $X = [0,2\pi]$ Assume $X$ has a uniform distribution. Let $f: X \rightarrow 1$ be a transformation of the $X$ to a random variable $Y$ whose only member is the number $1$ Can I ...
1
vote
1answer
114 views

$E (XY)=E (xE (Y|X)) $ - is this always true?

By the law of total expectation, $$E \left(XY\right) =E \left[E \left(XY\mid X=x\right)\right]=\dots$$ Now, because $X=x $ is a constant and by linearity of expectation, $$\dots=E [xE (Y|X=x)]$$ ...
2
votes
2answers
45 views

Probability of getting the same number three times.

If I have a set of numbers $\{1 \ldots n\}$ where $n \ge 1$ and I pick $3$ numbers from the set independently and uniformly. Whats the probability I'll get all $2$'s, the probability I get all the ...
2
votes
1answer
76 views

Does $\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)$ imply independence of $X$ and $Y$? [duplicate]

It shouldn't, but I am blanking on a counterexample. ETA: Note that the $t$ is shared on both sides - which differentiates this from this question. Similarly $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ implies ...
1
vote
1answer
42 views

Expected value for choosing number twice

I have a problem: I can choose twice randomly numbers from 1 to 3 and probabilities to choose each numbers can be adjusted for best end result. Choosing a number is independent. But not equally ...
0
votes
0answers
10 views

Pairwise independent hash functions vs. p-wise independent hash functions, what is the difference?

I am reading a paper and it mentions how pairwise independent hash function gives weaker, but still sufficient results, comparing to p-wise independent hash functions. I am not very familiar what ...
1
vote
0answers
21 views

How to show a Bayes Update is Normal Economics Example

I'm having trouble with an example of Bayesian updating in economics. The problem I'm trying to solve is from Sargent and Ljungvist's Recursive Macroeconomic Theory, Chapter 6. A firm wishes to ...
2
votes
3answers
244 views

Confusion about the range of the sum of i.i.d. random variables

Let $X_1, X_2, ...X_n$ be independent and uniformly distributed random variables on the interval $[0,1]$. Now suppose I wanted to calculate the probability density function of $Z = X_1 + X_2 + ... + ...
0
votes
1answer
45 views

Probability of a die

I am unsure about how to do this problem. A die has faces 1 to 6 and is weighted so that the probability of throwing $n$ in a single throw is proportional to $n$ ($n = 1,2,3,4,5,6$); that is, equals ...
0
votes
2answers
31 views

How to find probability in “exactly” situation type questions?

There are two boxes, each containing two components. Each component is defective with probability $1/4$, independent of all other components. The probability that exactly one box contains exactly one ...
0
votes
1answer
33 views

Integral of a distribution function

I am attempting to prove the following identity for the random variable R defined on ($ -\infty $, $ +\infty $). Upon attempting to integrate by parts I run into an indeterminate form. I am not sure ...
1
vote
2answers
93 views

What's the probability that we don't have $3$ consecutive heads in $n$ tosses?

In particular, I'm supposed to derive the following recurrence relation: $$Q_n = \frac{1}{2}Q_{n-1}+\frac{1}{4}Q_{n-2}+\frac{1}{8}Q_{n-3}$$ where $Q_0=Q_1=Q_2=1$ and $Q_n$ represents the probability ...
-1
votes
2answers
56 views

What is the probability that a committee of $4$ people chosen from $5$ married couples does not contain a husband and wife? [closed]

"A committee of four is chosen at random from 5 married couples. The chance that the committee will not include a husband and his wife is $A. 8/21$ $B. 6/17$ $C. 9/17$ $D. 5/21$" How to do$?$
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0answers
52 views

Expected value of sum of heights of books in a shelf with limited width

This question has arisen from a previous post: Statistical problem: how many books of different widths fit it into a self of a limited certain width? Let's assume that there are $N$ types of books, ...
-1
votes
1answer
22 views

What is the correlation between the pairwise differences of 2 bivariate normal random variables? [closed]

Given (X,Y) bivariate normal, $U = \frac{X_i - X_j}{\sqrt2\sigma_x}$ and similarly $V = \frac{Y_i - Y_j}{\sqrt2\sigma_y}$ for any two independent pairs $(X_i, Y_i)$ and $(X_j, Y_j)$. Why is this true ...
1
vote
2answers
35 views

Probability of a draft without replacement

There is an urn with $N_1$ balls of type $1$, $N_2$ of type $2$ and $N_3$ of type $3$. I want to show that the probability of picking a type $1$ ball before a type $2$ ball is $N_1/(N_1+N_2)$. ...
0
votes
1answer
30 views

Probability of $3$ questions

"A boy answers three questions. If one mark is given to each correct answer and the probability that he answered a question correctly is $0.6$, find the probability that he gets no marks." How to do ...
2
votes
3answers
115 views

Probability. White and black sheeps

Of these days I met an interesting theoretical probability problem. There is a flock of sheep, which initially included one white and one black sheep. Every day, the following procedure: choose ...
1
vote
0answers
27 views

Does $X \in L^p$ imply $\mathbb{E}[X|\mathscr{G}] \in L^p$?

Let $X$ be a real-valued random variable on a probability space $(\Omega, \mathscr{F}, P)$ such that $X \in L^p$ for some $p>1$. Given $\mathscr{G}\subseteq \mathscr{F}$, is it true that the ...
0
votes
1answer
43 views

In a certain state $55$ vote for Donald. $3$ voters are selected at random find the probability that…

In a certain state $55$ vote for Donald. $3$ voters are selected at random find the probability that A) all 3 voted for donald B) exactly 2 will vote for donald C) all 3 will not vote for donald ...
0
votes
1answer
44 views

How is the equation $P(X_1>X_2) = \displaystyle \int_{0}^1 P(X_1>X_2 | X_2=x) f_{X_2}(x) dx$ derived?

In Probability and uniform distribution, the following equation is used: $P(X_1>X_2) = \displaystyle \int_{0}^1 P(X_1>X_2 | X_2=x) f_{X_2}(x) dx$ How the equation is derived? Base on which ...
0
votes
6answers
57 views

Probability of urns

Four identical urns each contain 3 balls. In urn one, all three balls are black; urn two, 2 black 1 white; urn three, 1 black and 2 are white; urn four, all balls are whites. One of the urn is picked ...
0
votes
2answers
36 views

Conditional Probability of marbles

$ 1.$ Box A has $4$ green marbles and $5$ red marbles. Box B has $3$ green marbles and $2$ red marbles. A Marble is selected at random from box B and is put into box A. Then a marble is selected from ...
3
votes
2answers
80 views

the ant walk (year $12$) [closed]

An ant stands in the middle of a circle ($3$ metres in diameter) and walks in a straight line at a random angle from 0 to 360 degrees. Problem is, it can only walk one metre before it needs a break, ...
3
votes
1answer
51 views

Explanation of example of Banach Tarski .

I am reading Probability with Martingales by David Williams. In the book he gives an example where Banach and Tarski showed that if the Axiom of Choice is assumed, as it is throughout conventional ...