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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0answers
6 views

Probability between 3 parties

i have a question as below: probability of A win over C is 0.1 (probability of A lose to C is 0.9) and probability of B win over C is 0.4 (probability of B lose to C is 0.6). What is the probability ...
2
votes
2answers
24 views

Prove that $\mathbb P(X>Y) =\frac{b}{a + b}$ if $X, Y$ are exponentially distributed with parameters $a$ and $b$.

Let $X, Y$ be an exponentially distributed random variables with parameters $a, b$. Then $X$ has pdf: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
1
vote
2answers
17 views

Meeting probability of two bankers: Uniform distribution puzzle

Two bankers each arrive at the station at some random time between 5PM and 6PM (arrival time for each of them is uniformly distributed). They stay exactly five minutes and then leave. What is the ...
0
votes
1answer
9 views

$n$ points be placed uniformly at random on the boundary of a circle of circumference $1$.. what's the $n$ arcs' length distribution?

Let $n$ points be placed uniformly at random on the boundary of a circle of circumference $1$. These n points divide the circle into $n$ arcs. Let $Z_i$ for $1 \le i \le n$ be the length of these ...
0
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3answers
7k views

Discrete Probability Problem: determining probability mass function and cumulative distribution function

Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is .4 (a ...
0
votes
1answer
25 views

Given sum of uniform random variables $Z_1 + Z_2 + \dots + Z_n =1$,what's the probability that $k$ R.Vs are at least $1/n$?

Given sum of uniform random variables on $[0,1]$, $Z_1 + Z_2 + \dots + Z_n = 1$, what is the probability that exactly k random variables are at least $\frac{1}{n}$? In other words, what's ...
1
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0answers
9 views

Esscher Transform extended

The Esscher-transform is a well know tool in the financial section. I posted this in statistics also, since it relates to continuous sampling. Im not sure if my approach is right, so it would be nice, ...
2
votes
0answers
325 views
+50

Create the most 'stressful' tennis game ever!

Some games, such as tennis, use a complicated points system (point, game, set, match; with deuces and tie-breaks) for what would otherwise be an extremely simple and monotonous game. The main reason, ...
2
votes
1answer
25 views

Conditional Radon-Nikodym and disintegration

Here (p. 15) the author defines conditional divergence as $$D(P_{Y\mid X}\mid\mid Q_{Y\mid X}\mid P_X):=\mathbb{E}_{x\sim P_X}\left[D(P_{Y\mid X=x}\mid\mid Q_{Y\mid X=x})\right]$$ for two ...
0
votes
0answers
14 views

Deriving the Kalman filter using basic facts about normally distributed random variables

I'm attempting to derive the Kalman filter using basic facts about normally distributed random variables. Can anyone complete this derivation? Here's what I have so far (there could be some ...
0
votes
1answer
10 views

Given a probability generating function what is the $r$th term

Given that pgf is $G(H(\xi))=\frac{1+\xi}{3-\xi}$. Where $H(\xi)=\frac{1}{2}(1+\xi)$ and $G(\xi) = \frac{\xi}{2-\xi}$. And that $G(H(\xi))$ is the pgf of some random variable $Y$. How does one get the ...
8
votes
1answer
504 views

If two sets have a natural (asymptotic) density, does their union?

Let $\Omega=\mathbb{N}$. For each $E\subset\Omega$ let $N_n(E)$ be the cardinality of the set $E\cap [1,2,\ldots,n]$. Define $C=\left\{E: \lim_{n\rightarrow \infty} \frac{N_n(E)}{n} \text{ ...
3
votes
1answer
30 views

Using the Weak Law of Large Numbers for a product or random variables?

I need to calculate the average of the following quantity: \begin{equation} S_n=\prod_{i=1}^nS(X_i) \tag{1} \label{eq:1} \end{equation} with $S(X_i):=o_{X_i}b_{X_i}$, where each $X_i\in ...
0
votes
0answers
22 views

Find continuous stochastic variable $X$ with PDF $f_X = \frac{1}{x^2}$

Given the uniform stochastic variable $U$ defined on the interval [0,1]. Using $U$, define a continuous stochastic variable $X$ with probability density function (PDF) $$f_X(x) = \begin{cases} ...
0
votes
0answers
15 views

Why is the probability of extinction given by the probability generating function applied to 0?

I am trying to understand branching processes and can't find a good explanation for why solving for the probability of extinction at time $n$ is given by $p^{(n)}(0)$ with the superscript ...
0
votes
0answers
22 views

Tail bounds for functions of a Poisson point process

A Poisson point process consists of a sequence of points $0\leq t_1\leq t_2<\cdots$ where $t_i = t_{i-1} + X_i$ where $X_i$ is an exponentially distributed random variable with some rate parameter ...
5
votes
5answers
886 views

If two coins are flipped and one gets head, what is the probability that both get head?

I have a doubt because I think that once the result of the first coin is obtained, just simply await the outcome of the second, which is completely independent of the previous one, and then we have a ...
1
vote
3answers
26 views

Candies withdrawal probability for a particular subsequence

You are taking out candies one by one from a jar that has 10 red candies, 20 blue candies, and 30 green candies in it. What is the probability that there are at least 1 blue candy and 1 green candy ...
-3
votes
1answer
27 views

what is the probablility of the event occuring? [on hold]

If there is a 1 in a million chance of something happening, in million attempts the likelihood of it happening is a) 50% b) 63%
0
votes
2answers
42 views

Interpretation of correlation (coefficient)

In an discussion we were confronted with a very special opinion about correlation in respect of financial assets. The widely used correlation coefficient is used here to give an idea about how ...
2
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0answers
43 views

How to generate correlated random numbers with specific distributions?

After read the answers of some similar questions on this site, e.g., Generate Correlated Normal Random Variables Generate correlated random numbers precisely I wonder whether such approaches can ...
1
vote
5answers
4k views

Sum of total scores odds will end even or odd

Friend and i are betting on the ending total scores of basketball game, he says odds are greater it will end with an even sum, i say the odds are equal. Who is right?
0
votes
1answer
20 views

Formula for probability of being $\epsilon$ within the mean.

It should be possible to restate that as $P(\mu-\sigma \Phi^{-1}(\frac{p+1}{2})\leq X\leq \mu+\sigma \Phi^{-1}(\frac{p+1}{2}))=p$. In this answer, it says: For a normal distribution, the ...
1
vote
3answers
57 views

First to the sequence HT between two players

Two players, A and B, alternatively toss a fair coin (A tosses first and then B). The sequence of heads and tails is recorded. If there is a head followed by a tail (HT subsequence), the game ends and ...
0
votes
0answers
56 views

Probabilities in this blackjack variation

Let's say I play blackjack (52 cards, figures count for 10, aces count for 1 or 11) and alone (no dealer). The cards I use for one particular game are always removed at the end of that game and won't ...
-2
votes
0answers
26 views

¿Could anyone give me an example of the rice distribution? [on hold]

I'm studying the rayleigh and rician distribution, but i need an example of rician pdf, an application of the function in real life if you can explain it step by step
1
vote
1answer
33 views

Is this a correct interpretation of maximum likelihood estimation?

Here is an excerpt from Pattern Recognition and Machine Learning by Christopher Bishop: This seems to be not quite right—"the probability of the data set", when the data set is drawn from a ...
0
votes
0answers
23 views

The distributon function for the duration of a certain soap opera (in tens of hours) is F(x)=1-(16/x^2) , x> or equal 4 and F(x)=0 x<4

The distributon function for the duration of a certain soap opera (in tens of hours) is F(x)=1-(16/x^2) , x> or equal 4 and F(x)=0 x<4 a)calculate f the probability density functon of the soap ...
4
votes
3answers
33 views

Birthday line to get ticket in a unique setup

At a movie theater, the whimsical manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket. You ...
-2
votes
1answer
25 views

Let $f(x)=2 x^{-3}$ for x between 1 and Infinity, $f(x)=0$ otherwise be the pdf for a random variable $X$, find $F(x)$ [on hold]

Let $f(x)=2 x^{-3}$ for x between 1 and Infinity, $f(x)=0$ otherwise be the pdf for a random variable $X$, find $F(x)$. Could you please help me how can I find it ?
3
votes
2answers
1k views

Proof of Frechet-Hoeffding Copula bounds

How is the lower Frechet-Hoeffding copula bound proved? In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$. I'm struggling to ...
1
vote
1answer
17 views

let $f(x)=(3(x+x^2))/14$ and $x$ between $0$ and $2$ , zero otherwise be the pdf for a random variable $X$ ,Find the median and the mode?

let f(x)=(3(x+x^2))/14 and x between 0 and 2 , zero otherwise be the pdf for a random variable X Find the median and the mode ` Could you please help me Is it correct or not?
-1
votes
1answer
25 views

Poisson random variables [on hold]

A and B are two independent Poisson random variables The number of arrivals of A is x per hour The number of arrivals of B is y per hour E(A) = 12, E(B) = 7 (these are expected values). If there are ...
-1
votes
2answers
39 views

Probability density function for product and minimum of i.i.d. $U(0,1)$ random variables

If $U$ and $Y$ and $Z$ are i.i.d. $U(0,1)$ random variables, find the pdf for $A= U \times Y$ and $B = \min \{ U,Y,Z\}$.
1
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0answers
40 views

Probability generating functions of coin tosses

I have just came across a weird definition for the probability generating function of a random variable $N$ that denotes the integer value for the $n^{\mathrm{th}}$ toss on which the coin turned out ...
2
votes
1answer
15 views

Probability of miscommunication (bits)

I am struggling with the following problem from Blitzstein-Introduction to Probability: Alice is trying to communicate with Bob by sending a message across a channel. a). First she sends only one ...
1
vote
1answer
53 views

Sharing pears and bananas

Per has $3$ bananas and $5$ pears. Olav asks if he could have some fruit and Per agrees. What is the probability that he receives half ($1/2$) a pear and three quarters ($3/4$) of a banana? ...
1
vote
2answers
35 views

Coin Toss Game - Probability of H when unequal number of coins tossed

Two gamblers are playing coin toss game: Gambler A has (n+1) coins and B has n coins. What is the probability that A will have more heads than B if both flip all their coins. Not sure how to go about ...
3
votes
1answer
36 views

Is $\{\frac1n\sum_{k=1}^n X_k\ \text{converges}\}$ a tail event?

Suppose that $X_1,X_2,\dots$ is a sequence of random variables on some probability space. The tail $\sigma$-algebra $\mathcal{T}$ is defined as the intersection of $\sigma$-algebras ...
0
votes
1answer
528 views

Conditional Probability teenage drivers

Teenage drivers pay more for automobile insurance than older drivers. Many companies offer discounts for teenage drivers good grades. Assume that 20% of all teenage drivers are involved in accidents ...
2
votes
1answer
600 views

Approximate th Probability of a Sum of 16 Independent Uniform R.V.s

This question has to do with the Central Limit Theorem, uniform random variables, and cumulative distribution functions, I believe, but I'm not quite sure how to apply them all in the proper way. Q: ...
1
vote
1answer
568 views

Binomial Distribution for defects

I'm stuck on the following problem: A batch of components has arrived at a distributor. The batch can be characterized only if the proportion of defective components is at most 0.10. ...
171
votes
17answers
30k views

Do men or women have more brothers?

Do men or women have more brothers? I think women have more as no man can be his own brother. But how one can prove it rigorously? I am going to suggest some reasonable background assumptions: ...
2
votes
1answer
34 views

Expected values of $\max(X,Y)$ and $\min(X,Y)$ for $N(\mu,\sigma^2)$ distributed $X$ and $Y$

Suppose that $X$ and $Y$ are independent and $N(\mu,\sigma^2)$ distributed. Then $E(\min(X,Y))=\mu-\frac{\sigma}{\sqrt{\pi}}$ and $E(\max(X,Y))=\mu+\frac{\sigma}{\sqrt{\pi}}$. I tried to ...
1
vote
1answer
29 views

how many numbers drawn more than once

There are 100 numbered balls in an urn. We make 100 random draws with replacement. Of course, we can not expect to draw every number exactly once, there will be multiples. What is the expected value ...
0
votes
1answer
32 views

Let $X$ be a random variable with mean $0$ and finite variance $\sigma^2$. By applying Markov’s inequality show that

I am looking for confirmation that I am working in the correct direction as well as pointers for points where I have gone astray. Here is the problem. (a) Let $X$ be a random variable with mean $0$ ...
0
votes
0answers
31 views

CDF to PDF - Piecewise

Consider a random variable $X$ having the following PDF $$f(x)=\begin{cases}c,&\text{for }0<x<2\\2c,&\text{for }5<x<10\\0,&\text{otherwise}\end{cases}$$ ...
5
votes
1answer
47 views

Probabilities ant cube

I have attached a picture of the cube in the question. An ant moves along the edges of the cube always starting at $A$ and never repeating an edge. This defines a trail of edges. For example, ...
0
votes
1answer
7k views

General Addition Rule for Probability extended to 4 events?

I just started statistics and need to use the general addition rule. I know what it looks like for $3$ events: $$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) - (2 ...
1
vote
1answer
271 views

Two different sequences of random variables each converge in distribution; does their sum?

My question is about basic probability. We have two sequences of random variables, $ \{ X_n \}$ and $\{ Y_n \}$, such that each converge in distribution - i.e. there exist random variables $X$ and ...