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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2answers
13 views

Probability problem involving waiting time at pharmacy

I'm working on the following problem: The case of Safeway is very easy as the answer is simply the mean of $f_T$, namely $1$ minute. However, I'm having problems solving this problem for Target. ...
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1answer
11 views

Probability of random assignment to form pairs

So the question goes: I have 100 individuals and 100 different buses, and I randomly assigned each individual to sit on a bus (each bus has equal probability of being selected). How many buses are ...
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2answers
16 views

Book Problem Probability

A shelf has 2 math books and 3 physics books. Two of the books are selected at random. Let X be the number of math books in the sample. Construct a probability table for X. Find E(X) using the table. ...
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2answers
19 views

expected number of moves to match pairs in matching game

There are 6 tiles made up of 3 matching pairs (e.g. 1,1,2,2,3,3). They are mixed up and flipped over so you can't see the numbers. Assume you have perfect recall (i.e. if you flip over a tile, you ...
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0answers
12 views

Problem with constructing a uniform probability measure over $\mathbb{N_0}$ using rationals on the unit interval

I've been toying with the possibility of constructing a uniform probability measure over $\mathbb{N^0}$. Obviously, one cannot just assign each non-negative integer a probability of 0 and call it a ...
0
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1answer
29 views

Expected value and Variance calculation

Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are ...
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3answers
103 views

Alternative Monty Hall Problem

So the typical set up for Monty Hall problem, there are 3 doors where 2 have goats and 1 has a car. I, the contestant, get to randomly guess a door looking to get the one with the car, after this the ...
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3answers
17 views

Expectation of nonnegative Random Variable [duplicate]

Can someone help me give me some pointers as to how to prove this relation?
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0answers
20 views

statistics problem, where did I mistake?

I searched interesting problem about statistic from http://www.mast.queensu.ca/~stat353/resources/pastfinals/final12sol.pdf $$ $$ But at the question No.2, I have some problem the red box $$ $$ ...
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1answer
12 views

Finding expectation from joint PDF

Consider the following joint PDF for random variables $X$ and $Y$: (the height that the shading going up to on the $y$-axis is $0.5$, it just didn't show up for some reason). I'm trying to find ...
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1answer
32 views

Calculating the expected winner of a Penney's Game using a Markov Chain.

I am trying to calculate the probability that one sequence of coin tosses is more likely to win than the other in a game of Penney's. The sequences are: HTHT and THTT. So far I've come up with the ...
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0answers
15 views

Probability for matching hexagons number.

I have some interesting math problem. Let say we have nnumber of hexagons. Each hexagon has random numbers in his corners from ...
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0answers
13 views

maximum likelihood of a dirichlet prior

Suppose $\theta \sim D(\alpha)$ where $D$ denotes the Dirichlet distribution and $\alpha = (\alpha_1,\ldots,\alpha_K)$ its hyperparameter, in which case: $$p(\theta) = \frac{\Gamma(\sum_k ...
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2answers
15 views

What does it mean for a random variable to “admit” a distribution?

Can someone explain the word "admit" and explain what would happen if it does not admit a distribution?
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2answers
16 views

The Chernoff bound for continous rendom variables.

In a paper I am reading, authors apply the Chernoff bound to a continuous random variable $X$ with positive mean: $$\mathbb{P}(X\le 0)\le \mathbb{E}[\exp(\lambda X)]$$ I do not understand it. When I ...
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votes
2answers
10 views

Block Problem w w/o Replacement

Assume that a bin has 40 green and 60 red blocks. Four are selected at random. Let X be the number of green blocks in the sample. Construct the probability table for X assuming that sampling is: a. ...
0
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1answer
14 views

Let a random experiment be the cast of a pair of unbiased 6-sided dice and let X=the smaller of the outcomes?

Let a random experiment be the cast of a pair of unbiased 6-sided dice and let X=the smaller of the outcomes? a. With reasonable assumptions find the p.m.f. of X. b. Let Y equal the range of the two ...
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0answers
9 views

Proving that the Poisson random variable with mean $x$ is the Poisson random variable $X$ with maximum value of $\text{Pr}(X=x)$.

For any pair of positive numbers $\mu_1$ and $\mu_2$, let $X_1$ be the Poisson random variable with mean $\mu_1$, and $X_2$ be the Poisson random variable with mean $\mu_2$. Proof that $\text{Pr}(X_1 ...
0
votes
1answer
16 views

Number of bits needed for Huffman code

Jake uses a Huffman code to compress i.i.d. (independent nad identically distributed) strings of symbols that come from a 5-ary alphabet ($A$, $B$, $E$, $R$, $S$) where the probabilities of occurrence ...
1
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4answers
4k views

How to prove if P(B|A) > P(B) then P(B|A') < P(B)

I hate proof questions, I never know how to start. My question says: "Let A and B be two events in a sample space such that 0 < P(A) < 1. Let A' denote the compliment event of A. Show that if ...
1
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1answer
11 views

Probability mass function of a degenerate distribution

Wikipedia article "degenerate distribution" states that "The probability mass function does not exist." Is it really right? Why can't it be set as $$ f(x) = \begin{cases} 1 & \quad x = x_0, \\ ...
0
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2answers
39 views

The water heater problem ( mathematician or plumber)??

Isn't it absurd, I mean doesn't it make probability absurd. $\textbf{Problem-}$ Suppose my water heater broke and heat in my apartment raised high. I went to a "person" to ask him to take a look at ...
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1answer
216 views

finding unconditional distribution by integrating conditional distribution

Given $$ f_Y (y)= \begin{cases}\frac{1}{120} e^{-\frac{1}{120}y} &, y\ge 0 \\ 0, &, y< 0 \end{cases}$$ and $$f_{X|Y} (x|y) = \begin{cases}\frac{1}{y} &, x\in [0, y] \\0 &, ...
4
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2answers
822 views

Probability distribution function that does not have a density function

What is an example of the probability distribution function that does not have a density function?
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1answer
15 views

Ratio of hazards in Proportional Odds model

In the proportional odds model we have the the odds of survival in 1 group are proportional to the odds of survival in another group $$\dfrac{ S_1(t)}{1-S_1(t)} = \psi \dfrac{S_0(t)}{1-S_0(t)}$$ ...
1
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0answers
16 views

Hypergeometric Distribution2

From a class of 10 boys and 15 girls, prizes are randomly awarded to 3 children. Let N be the number of boys who win prizes. Construct the probability tables for the random variable N assuming that ...
15
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5answers
1k views

Probability that the sum of three integer numbers (from 1 to 100) is more than 100

I have an urn with $100$ balls. Each ball has a number in it, from $1$ to $100$. I take three balls from the urn without putting the balls again in the urn. I sum the three numbers obtained. What's ...
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0answers
12 views

Kurtosis Inequality

I´m trying to proof the following: Proof that $Kur(X)$ $\geq-2$ with equality if and only if $\mathrm{P(X=1)=1/2}$ and $\mathrm{P(X=-1)=1/2}$. Where $Kur(X)$ is the Kurtosis i.e. $Kur(X) = ...
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1answer
32 views

Probability rolling die question [on hold]

Suppose we have three fair 6-sided die (one colored green, one colored black, and one colored red). An experiment consists of rolling all three die. Let G, B, and R denote the numbers rolled on the ...
1
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0answers
15 views

Showing $E[e^{-\lambda \tau_{a}\wedge\tau_{-a}}]=sech(a\sqrt{2\lambda})$

This is homework so no answers please. For $\tau_{a}=inf_{t}(B_{t}=a)$ , we already know $E[e^{-\lambda \tau_{a}}]=e^{-\sqrt{2\lambda}a}$. By $B_{t}$ I mean Brownian motion. The question is to show: ...
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2answers
29 views

Probability question with variables [on hold]

Jane has two children which were born on different dates. In the community that Jane is from (and hence you can assume this is also true for Jane) the possibilities when one has two children are {(b, ...
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2answers
19 views

probability density function question

The diameter of grains of sand from a sand pit, measured in mm, can be considered a continuous random variable X with probability density (picewise function) f(x)=4(x-x^3), 0<=x<=1, otherwise ...
2
votes
1answer
28 views

Elevator Wait Time

Question: A building contains one elevator which can access each floor numbered $0$ through $m>0$. It picks up $n\le m$ passengers in the lobby (floor 0). If it takes the elevator $\alpha$ seconds ...
4
votes
2answers
129 views

Probability of a minimum sum of card values when drawing cards from a custom deck

Apologies if this question has been answered -- I tried searching for an answer but wasn't sure what terminology to use. Suppose I have a deck of cards, consisting of $n_1$ ones, $n_2$ twos, etc. ...
1
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2answers
62 views

$X_n \overset{a.s.}{\longrightarrow} X$ and $X_n \overset{L^1}{\longrightarrow} Y$ implies $X = Y$ a.s.?

If I have a sequence of random variables $\{X_n\}_{n \geq 0}$ such that $$X_n \overset{a.s.}{\longrightarrow} X \quad\textrm{and}\quad X_n \overset{L^1}{\longrightarrow} Y$$ then is it always true ...
2
votes
2answers
21 views

Bivariate distribution with normal conditions

Define the joint pdf of $(X,Y)$ as: $$f(x,y)\propto \exp(-1/2[Ax^2y^2+x^2+y^2-2Bxy-2Cx-Dy]),$$ where $A,B,C,D$ are constants. Show that the distribution of $X\mid Y=y$ is normal with mean ...
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0answers
17 views

Generalization of the Glivenko-Cantelli Theorem

The classic Glivenko-Cantelli Theorem states that $$ \sup_{t}|F_{n}(t) - F(t)| \longrightarrow_{a.s.} 0 $$ where $F_{n}(t)$ is the empirical cdf. Looking at the proof of the theorem, it seems to me ...
4
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2answers
114 views

Problems in elementary number theory and methods from physics (and other branches)

Yesterday I started wondering if there are intuitive "physical" arguments problems to solve problems from number theory (elementary number theory in particular, but also advanced topics) (for example, ...
1
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1answer
15 views

IQR of continuous random variable

If the continuous random variable $X$ has probability density $$ f(x) = \begin{cases} \sec^2 x, & 0<x<\dfrac\pi4, \\[8pt] 0, & \text{otherwise,} \end{cases} $$ find the interquartile ...
0
votes
1answer
36 views

Random walk question

here is the problem that I have been trying to do: N+1 plates are laid out around a circular dining table, and a hot cake is passed between them in the manner of a symmetric random walk: each time it ...
0
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2answers
42 views

Probability paradox: Mario's dice game

Consider the following: Mario invites an infinite number of friends to a party, and challenges them at a game. First, Mario randomly splits his infinite friends in groups of incremental size, the ...
2
votes
1answer
26 views

Finding $\mathbb{E}(X+1)$ and $\mathrm{var}(X+1)$ of a Poisson rnd variable

In this exercise: Let $X$ be a Poisson random variable with parameter $\lambda$ and let $Y=X+1$. Find $\mathbb{E}(Y)$ and $\mathrm{var}(Y)$. I was able to apply the definition of expected value ...
0
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0answers
9 views

Concentration of the absolute norm of a normal distribution given via this integral?

Let $\mu\sim N(0,\Sigma)$ be a multivariate Guassian distribution. What is the natural meaning of the integral: $\int_{[-a_\alpha,a_\alpha]} x^2 \bar{\mu}(dx)$? This is a partial second moment, ...
2
votes
1answer
76 views
+50

Books that use probabilistic/combinatorial/graph theoretical/physical/geometrical methods to solve problems from other branches of mathematics

I am searching for some books that describe useful, interesting, not-so-common, (possibly) intuitive and non-standard methods (see note *) for approaching problems and interpreting theorems and ...
0
votes
1answer
42 views

Is there a result that the density function for $\chi^{2}$ must be related to the standard normal density?

Suppose we have a random variable $X$ with the property that $-X = X$ (in distribution) and $X^{2} = \chi^{2}(1)$? I want to be able to conclude that $X \sim N(0,1)$. The probability density function ...
1
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2answers
46 views

Present a combinatorial argument for the identiy $\sum^{n}_{k=1} k\binom{n}{k} = n\cdot 2^{n-1}$

This is a question in my textbook that does not provide a solution. Any help on a solution? Consider the following identity: $\sum^{n}_{k=1} k\binom{n}{k} = n\cdot 2^{n-1}$ Present a combinatorial ...
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0answers
22 views

Number theory: Prove that random variables are independent

We consider $\mathbb{Z}_p=\{0,\dotsc,p-1\}$ for a prime $p$. Now we choose $a$ and $b$ uniformly at random from $\mathbb{Z}_p$ and define $X_{i,r}$ as the indicator random variable for the event that ...
0
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1answer
17 views

How to calculate the expected number of times a specific pattern appears in a set of n numbers of Bernouilli trials?

Let's consider a set of n Bernouilli trials of parameter p $\text{(O1,...,On)}$. Let's name the two possible outcomes as ...
0
votes
1answer
23 views

How logarithms affect given condition

I am working with long productcs of probabilies and in order of avoinding underflow I am using the addition of (negative) logarithms. P(A) =-log(P(a1) + -log(P(a2)+.... In the end I get a positiv ...
5
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8answers
7k views

Deal or no deal: does one switch (to avoid a goat)?/ Should deal or no deal be 10 minutes shorter?

Okay so this question reminded me of one my brother asked me a while back about the hit day-time novelty-worn-off-now snoozathon Deal or no deal. For the uninitiated: In playing deal or no deal, the ...