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
73 views

Uniform convergence in integrated survival function implies uniform convergence of distribution functions?

For a probability distribution function $F$ supported on a bounded interval $[a,b]$, the integrated survival function (ISF) is defined as $$\Psi_F(t)=\mathbb E_F\max\{X-t,0\}=\int_t^b(1-F(x))d x.$$ ...
2
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0answers
160 views

Comparing the stopping times of two stochastic processes

Let $f_1$, $f_0$, $g_1$, $g_0$ be $4$ distinct density functions on the real numbers $\mathbb{R}$ with the corresponding distribution functions $F_1$, $F_0$, $G_1$, and $G_0$, respectively. The ...
3
votes
1answer
363 views

Square of Bernoulli Random Variable

I was wondering about the distribution of the square of a Bernoulli RV. My background in statistics is not too good, so maybe this doesn't even make sense, or it is a trivial problem. Let, $Z\sim ...
1
vote
1answer
51 views

expected value and negative binomial distribution

Repeatedly roll a fair die until the outcome 3 has occurred on the 4th roll. Let X be the number of times needed in order to achieve this goal. Find E(X). My Attempt : Pr(getting a 3 on the 4th ...
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4answers
293 views

Probability Exercise - Contracts

Here I have this exercise which I am not sure of how to approach, it is in the conditional probability section but I cannot see the use in here, I will state the question and then state my intuitive ...
0
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1answer
47 views

Probability, Linear Models, Expectation

I'm trying to find a way of predicting various models from one "perfect model", using EXCEL. i.e. If I assume that all models should behave like the original one, for which I have all the ...
1
vote
1answer
75 views

Conditional Independence

Consider three r.v. X,Y and Z. The r.v. X is independent of Z given Y (i.e. $X \perp Z \ | \ Y$) then the following is true (for some reason): $Pr[x,y,z] = \frac{Pr[x,y]Pr[y,z]}{Pr[y]} = ...
0
votes
1answer
321 views

Probability of X heads before Y consecutive tails in N biased coin tosses

I have another coin toss question: Assume I am tossing a biased coin n times with probability p of coming up heads. What is the probability that x heads come up, before y consecutive tails? A code ...
0
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2answers
159 views

Expectations of functions of normal random variables

so I am a TA in an intro stats class and I stumbled upon a brain teaser question that even I am not quite sure how to solve. I thought some of you might be able to help. The question is as follows: ...
0
votes
1answer
126 views

Mean and Variance

Thirty-seven in one hundred travellers named “crying kids” as the most annoying on a flight. You randomly select eight people and asked them if crying kids are the most annoying on a flight. Compute ...
4
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1answer
136 views

Asymptotics of the classical occupancy problem

Classical Occupancy Problem. There are $n$ distinct labeled balls in an urn. $k$ of them of uniformly selected with replacement. What is the probability that the sample contains at least one ball ...
0
votes
1answer
274 views

Average and expected value in a biased coin toss

Not sure on which SE site to ask this. It is essentially a math question however I am looking for a practical code solution. I have a loop with an integer i which changes (according to a predefined ...
0
votes
2answers
432 views

Probability of getting at least one number out of 10 with 4 chances

Let's say that in a lottery, 4 balls are drawn at random from a pot of 10 balls, numbered from 0 to 9. The drawn balls are not put back in the pot. If your ticket matches the 4 drawn balls you win the ...
4
votes
2answers
67 views

Probability question - how many cycles before all items are chosen

I have a container of 100 yellow items. I choose 2 at random and paint each of them blue. I return the items to the container. If I repeat this process, on average how many cycles will I make ...
1
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2answers
220 views

Probability Exercise (Java and C++)

So I have this probability exercise and I'd like to know if it is correct, along with my reasoning, so here is the exercise: In a computer installation, 200 programs are written each week, 120 in ...
4
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3answers
259 views

Probability that a chosen number will be a Fibonacci number

Suppose that I randomly choose an integer $x$ with $1 \leq x \leq n$ where $n$ is a natural number. What is the probability that $x$ will be a Fibonacci number?
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2answers
46 views

Probability Theorem

So I have this theorem which I understand but proving it is more than weird to me. So it is: If $E_1$ and $E_2$ are subsets of $S$ and $E_1\subseteq E_2$ then $P(E_1)\le P(E_2)$ Which I ...
2
votes
3answers
156 views

I flip M coins, my opponent flips N coins. Who has more heads wins. Is there a closed form for probability?

In this game, I flip M fair coins and my opponent flips N coins. If I get more heads from my coins than my opponent, I win, otherwise I lose. I wish to know the probability that I win the game. I ...
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2answers
136 views

Too stupid to understand random variable questions?

I have two excercises: 1.) Let $X_1,X_2,X_3$ be independent uniformly distributed random variables on $[0,1]$. What is the density function of $X_1+X_2+X_3$? 2.) Let $X_1,...,X_4$ be independet ...
1
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2answers
109 views

Probabilistic “proof” that a sentence is provable (proof “density”).

Is it possible to (or even useful) to calculate the probability that a certain statement is provable? I had this idea that any two statements say A and B could be compared to each other by comparing ...
2
votes
2answers
297 views

Probability In Multi-Throw Dice Game

I am having some trouble coming up with a probability table for a game that I wrote. The game uses 6 dice. The player can throw the dice multiple times, but must retain at least 1 on each throw. In ...
0
votes
2answers
60 views

Probability of a linear combination of 4 independent normal distributed variables

I'm calculating the following probability: $P = P(F(a,b,c,d)<0)$ Where $F(a,b,c,d)$: $F(a,b,c,d)= 1500000 - 500a - 500000b - 100000c + 5000000d$ $a, b, c$ and $d$ are independent normally ...
1
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1answer
57 views

Question on geometric distribution

Suppose that the probability for an applicant to get a job offer after an interview is 0.01. An applicant plans to keep trying out for more interviews until she gets offered. Assume outcomes of ...
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2answers
65 views

what is meaning of “independent values of a random variable”?

I need some help with basic statistics terminology. Could someone please explain in layman terms the meaning of "independent values" regarding a random variable? Perhaps a six-sided die (with sides ...
4
votes
1answer
106 views

Find the probability the same color was used twice in a chess game given the player did not lose

Here's the question and its solution: I don't see how the solution to the problem is to compute: $[1-P(W|L)]^2+[1-P(B|L)]^2$ i.e. I don't think the expression above reflects what the question is ...
3
votes
1answer
524 views

Is this infinite sum always less than zero?(+500pts bounty for the correct answer)

I wonder if the following infinite sum is always negative for all (finite) $A,d>0$ and $B<0$. Any counterexample also suffice. Here is the sum: $$\frac{\partial}{\partial d}\sum_{n=1}^\infty n ...
1
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0answers
76 views

An inequalities on order statistics

Let $X=(X_1,...,X_k)$ be a square integrable random vector in $\mathbb R^k$ and $X_{(j)}$ the $j$-th ordered value of $X$, i.e., $X_{(1)} \leq X_{(2)} \leq ... \leq X_{(k)}$. Prove (or disprove) that ...
3
votes
1answer
118 views

How to arrive at a specific formulation of the relative median deviation? Related to integration and statistics.

So my title is not very specific but here is the question in more detail. I am an economist currently working with this book: Frank Cowell - Measuring Inequality On page 25 a formulation of the ...
4
votes
1answer
73 views

To prove the independency of two random variables

Suppose two random variables $X_1$ and $X_2$ are of identical independent distribution, with the same PDF $f(x) = e^{-x}, \space x>0$. Now, we have $$Y_1=\min(X_1, X_2)$$ $$Y_2=\max(X_1, X_2)$$ ...
3
votes
4answers
83 views

Finding generating functions - how was this jump made?

I'm going through examples of probability-generating functions in a book and am confused by the following example: $$1+2s+4s^2+...=\sum_{n=0}^\infty (2s)^n=(1-2s)^{-1}$$ I understand the summation but ...
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3answers
314 views

uniform moment generating function at t=0

I have calculated the moment generating function for the uniform distribution as Mx(t) = ((e^(tb)-e^(ta))/t(b-a) However I know Mx(0)=1 but I can't get my head around how this is possible as if t=0, ...
0
votes
1answer
46 views

how to reformulate general markov property in discrete case

I read the wiki article on the markov property http://en.wikipedia.org/wiki/Markov_property#Definition and wondered how to work out this reformulation. It seems intuitively but I can not work it out. ...
5
votes
5answers
192 views

Please explain to me why the Expected Value is $ E[X] = \int_{-\infty}^{\infty} x f_X(x) dx $

For probability density functions (at least for the normal distribution and beta distribution) it holds that the expected value is given by $ E[X] = \int_{-\infty}^{\infty} x f_X(x) \, dx $. I have ...
2
votes
1answer
2k views

Probability the three points on a circle will be on the same semi-circle

Three points are chosen at random on a circle. What is the probability that they are on the same semi circle? If I have two portions $x$ and $y$, then $x+y= \pi r$...if the projected angles are $c_1$ ...
0
votes
1answer
107 views

Compute probability of next coin toss

Given is a dataset with observations $\{h, t, t, h, t, t, t, t, t, h\}$. To compute the posterior probability function $f$ I assume a uniform Beta Distribution B(1,1) $ p(\theta) = B(1,1) ...
0
votes
1answer
19 views

About the variance and a connected integral

Is given a positive measure $\mu$ such that $\mu(\mathbb{R}^+)<+\infty$. Is it generally true that: $$\int_0^\infty x^2 d \mu < + \infty \space\space ^{?}\iff^{?} \int_0^\infty \left(x ...
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0answers
69 views

Two independent renewal processes

We have two urns (blue and red) that are connected, and two particles, $p_1$ and $p_2$, are traveling between these urns independently. The amount of time $Z_1$ that $p_1$ spends in blue urn is iid ...
2
votes
0answers
64 views

Entropy of sum of random variables

Let $x_1,x_2,\dots,x_n$ by random variables which take the values $0$ or $1$ with $P(x_i = 1) = p_i$ and $P(x_i = 0) = 1-p_i$, where $0 \leq p_i \leq 1$ for $i=1,2,\dots, n$. Let $$X= \sum_{i=1}^n ...
2
votes
3answers
92 views

What is the probability that their sum is divisible by $5$? [closed]

If a person picks $2$ distinct numbers in $\{1,2,3,\dots,9,10\}$, what is the probability that their sum is divisible by $5$? Thanks in Advance. answers option are: a) 6/45 b) 13/90 c) 7/45 d) 1/6
0
votes
1answer
43 views

Techniques for integrating this function?

I'm working my way through a textbook on probability in which the following integral appears: $$F(y)=\int_1^\infty y^{n-1}\lambda^ne^{-\lambda y}\frac{1}{(n-1)!}dy-\int_1^\infty ...
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0answers
183 views

How to calculate expectation of product of random variables

Let $X\sim U(1,n)$ and $Y\sim U(1,n)$ be two discrete random variables that distributes uniformly between $1$ and $n$. (I also have a joint distribution table of $\mathbb{P}_{X,Y}(X,Y))$. I need to ...
0
votes
0answers
24 views

Estimate the population mean when random selection is not possible

Consider I have a jar with marbles labeled 0 and 1 in it. They're not well mixed so the possibility of obtaining a sample sized 1000 with mean 0.6 and another sample sized 1000 with mean 0.4 is not so ...
1
vote
1answer
76 views

What's wrong with this transformation of random variables?

Consider $n$ random variables $t_1$ through $t_n$ each of which is uniformly randomly chosen on $[0,1]$ and labelled left to right so that $0\leq t_1 \leq t_2 \ldots \leq t_n \leq 1$. We readily see ...
1
vote
1answer
62 views

Joint To Marginal Density : Can't figure it out.

Here goes the problem: Problem: Suppose $X$ and $Y$ have the joint density function: $f(X,Y) = c \sqrt{1 - x^2 - y^2}, \,\,\,\,\, x^2 + y^2 \leq 1$ Find $c$. ...
0
votes
2answers
221 views

Probability- significant difference between $(A' \cap B) $ & $(A' \cup B)$ or $(A \cap B')$ & $(A \cup B')$?

Venn Diagrams in general! I've honestly been struggling with these kinds of questions for HOURS, I still don't get it. I went on the internet, researched to find an answer but only stumbled upon a ...
2
votes
0answers
52 views

How to compute next expectation?

$x$ and $y$ are normal random variables, $x\in N(\mu_x, \sigma_x^2), y\in N(\mu_y, \sigma_y^2)$ How to compute next expression $$ \mathbb{E} \arg (e^{ix} + e^{iy}) $$ In English, what is expected ...
0
votes
1answer
3k views

How to compute moments of log normal distribution?

The computed moments of log normal distribution can be found here. How to compute them?
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1answer
105 views

Statistic Textbooks

What is a good textbook for introductions to continuous and discrete distributions? The one that my university offers is a thin scrap put together by the department. Could I get some recommendations ...
0
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1answer
52 views

$ \mathbb{E}\left[\big(\mathbb{E}[X|Y]\big)^2\right] \leq \mathbb{E}(X^2)$

I'm working on old qualifying exam problems and haven't been able to get this one yet. Let $X,Y$ be random variables with joint density function $f_{XY}$, $\mathbb{E}(X^2)<\infty$, and ...
0
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1answer
55 views

probability ratio question

From n men and n women one wants to select k male and k female candidates, to create either a committee or a ballot. In a ballot the members are fully ranked (first, second, ...); in a committee they ...