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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3answers
52 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 ...
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2answers
18 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 ...
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0answers
5 views

How do you derive $P(\mu-\sigma \Phi^{-1}(\frac{p+1}{2})\leq X\leq \mu+\sigma \Phi^{-1}(\frac{p+1}{2}))=p$?

I got it from this answer, it says: For a normal distribution, the probability of being within $\Phi^{-1}\left(\frac{p +1}{2}\right)$ standard deviations of the mean is $p$, where $\Phi^{-1}$ is ...
0
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0answers
54 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 ...
-1
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0answers
23 views

¿Could anyone give me an example of the rice distribution?

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
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1answer
31 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 ...
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0answers
11 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
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3answers
29 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 ...
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votes
1answer
22 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
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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 ...
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1answer
16 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?
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1answer
21 views

Poisson random variables

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 ...
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2answers
38 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 $B = \min \{ U,Y,Z\}$.
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0answers
32 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
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1answer
13 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 ...
0
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1answer
28 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 ...
1
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1answer
51 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? ...
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2answers
34 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
34 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 ...
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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
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1answer
597 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: ...
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0answers
12 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 ...
1
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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. ...
170
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
32 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
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1answer
28 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
28 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
30 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
46 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
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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
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1answer
270 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 ...
2
votes
1answer
28 views

Infinite Marbles in a Jar with Known Distribution

Let's say I have infinite number of marbles in a jar and $90\%$ of them are red and $10\%$ are green. If I pick $25$ out of the jars (with or without replacement probably doesn't matter because the ...
2
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0answers
12 views

Levy process measurable past

For a Levy-process $(X_t)_{t\geq 0}$ with stationary indepedent increments which is a markov process, we know that its law is defined by its one dimensional distribution. This is so because for its ...
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1answer
18 views

Logic of getting a full-house of cards.

Although I understand the correct solution of finding the total number of full houses in a 52-deck of cards (finding the number of ways of selecting the first value and then finding the amount of ways ...
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1answer
573 views

Greatest of three random variables

Assume that we have $3$ not equal random variables $(A, B, C)$. If we know that $$Pr(A>B)=x, \quad Pr(A>C)=y, \quad Pr(B>C)=z$$ What is $Pr(A$ is the greatest one)? I know that $Pr(A$ is ...
1
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1answer
19 views

From brownian bridge to brownian motion proof

Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) ...
0
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1answer
18 views

Running Query on Conditional Probability graph

I want to run the following query. P(Rain | WetGrass = True). What I know: Because we are given the child, Rain and Sprinkler are no longer conditionally independent. My first approach is to use ...
3
votes
1answer
35 views

Confusion related to calculating the probability distribution of a variable

I have this confusion related to calculating the probability distribution of a variable. If I have a variable $x_1$ which has a pdf $p(x_1)$.Lets assume that the distribution is gaussian with mean ...
0
votes
1answer
70 views

Integrate $\int x \frac{f'(x)}{f(x)} dx$

I need your advice in integrating $\int ln(f(x)) dx = \int x \frac{f'(x)}{f(x)} dx$, where $f(x)$ is a probability density function. So it is the same as $\int x \frac{F(x)}{f(x)} dx$. How can I ...
2
votes
3answers
45k views

Find $E(XY)$ assuming no independence with $E(X) = 4$, $E(Y) = 10$, $V(X) = 5$, $V(Y) = 3$, $V(X+Y) = 6$.

I am having trouble finding a way to solve this problem. I understand this would be simple if $X$ and $Y$ were independent however, I believe they are not since $V(X) + V(Y) \neq 8$. Find $E(XY)$ ...
3
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0answers
40 views

Hitting probabilities in a random walk on a graph

Consider a random walk $(X_n)$ on the graph below, where we jump from a given vertex to one of its adjacent vertices with equal probability. I want to find: the probability that we hit $A$ before ...
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1answer
39 views

5 red and 10 black balls in a bowl, with replacement

Problem A bowl contains $5$ red and $10$ black balls. A ball is picked randomly and the colour is noted. After every pick the ball is placed back, and an extra ball of the same color is added to the ...
2
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1answer
40 views

Probability of picking marbles from a bag with only the ratio of marbles given

Here is a question that is puzzling me: A bag contains a large number of marbles; the numbers of the red, blue and yellow marbles are in the ratio $3:4:5$. Four marbles are randomly drawn ...
1
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1answer
28 views

Chevalier de mere paradox with game with three dice

Chevalier de Mere asked Blaise Pascal why in a game with three dice the sum $11$ is more favorable than $12$, when both sums have exactly the same possible combinations: For $11$ we have $(5,5,1), ...
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0answers
22 views

Brownian motion hitting time [on hold]

Let $B(t)$ be a linear Brownian motion and $a,b>0$. Show that $P(B(t)=a+bt \text{ for some } t>0)=e^{-2ab}$
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1answer
17 views

Probability of selection of $D$ if $C$ is elected unanimously

Five persons $A,B,C,D,E$ are contesting in an election in which $3$ persons are to be selected. If $C$ is elected unanimously, then find the probability that $D$ gets selected. I am not able to ...
2
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1answer
46 views

Is Brownian motion on $[0,b]$ bounded?

Is Brownian motion on $[0,b]$ bounded? Or at least bounded with probability one. Since Brownian motion is continuous with probability $1$, I guess the answer is YES.
0
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0answers
22 views

On the derivation of the Cauchy Distribution

I am currently studying from this video lecture series and the professor here goes over the derivation for the Cauchy distribution. I am able to follow most of it except for one minor part. Part of ...
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2answers
592 views

Elevator Probability Question

There are four people in an elevator, four floors in the building, and each person exits at random. Find the probability that: a) all exit at different floors b) all exit at the same floor c) two ...
2
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1answer
25 views

Find the PDF of $Y= \sin{(\pi X)}$, where $X \sim U[0,1]$

Let $X\sim U_{(0,1)}$ and lets define $Y= \sin{(\pi X)}$. I want to get the pdf of $Y$. My attempt: Clearly, $y\in(-1,1)\Rightarrow 1-y^2\ge0$, so $$ F_Y(y)=\Bbb P(Y\le y)=\Bbb P\big(\sin{(\pi X)}\le ...