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
83 views

Show that the E(|X|) is finite.

Show that if $E(X^2)<\infty$ then $E(|X|)<\infty$. My try: In other word, if $$\int x^2f(x)dx<\infty\Rightarrow\int xf(x)dx<\infty$$ for continuous case which $\int f(x)dx=1$ or $$\sum ...
0
votes
2answers
48 views

Chance of extinction

In minecraft, after chopping down a tree the leaf blocks have a 1/20 chance of dropping a sapling after they decay. Model this simpler and suggest that the trees give between 0 and 3 (inclusive, ...
5
votes
2answers
118 views

Non-geometric way to calculate expected value of breaks?

In "50 Challenging Problems in Probability", question #43 is the following: "A bar is broken at random in two places. Find the average size of the smallest, of the middle-sized, and of the largest ...
0
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1answer
45 views

Combinatorics and conditional probability

Assuming seven standard dice are rolled, what is the probability their sum equals 17? Show a general approach to solving this problem analytically, using conditional probability, combinatorics, etc
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2answers
139 views

Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
3
votes
2answers
566 views

A and B play until one has 2 more points than the other…

Question A and B play until one has 2 more points than the other. Assuming that each point is independently won by A with probability p, what is the probability they will play a total of 2n points? ...
2
votes
2answers
162 views

Using Markov - Chain to find average and probability

Suppose a computer generate a random vector of n positions where each position appears on of the numbers from 1 to n. The generation is performed uniformly on the $n!$ possibilities. In the problem we ...
1
vote
1answer
59 views

4 Children: What is the probability that one of them has a birthday this month?

I want to know what the probability is given a family having 4 children, that one child has a birthday this month (at any given time) How would this be expressed as an equation? How do you account ...
1
vote
6answers
75 views

Distribution of a binomial variable squared

If I know $X$ is a binomial random variable, how can I find the distribution of $X$ squared (I know that $P(Y=y=x^2) = p(X=x)$ but does this distribution have a standard name)? In particular, how can ...
1
vote
1answer
426 views

Three dice are thrown. What is the probability the same number appears on exactly two of the three dice?

This is my solution, not sure if this is correct. Let D1, D2, D3 be the outcome of the three dice. Since "exactly 2 of the 3 dice" is mentioned, I have the following sets: ...
0
votes
1answer
27 views

Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
2
votes
1answer
81 views

Expected value of log of 1+ a squared Gaussian random variable

If $X$ is standard normal, what is $$\mathbb{E}\log(1+X^2).$$ I see that $X^2\sim\mathrm{Gamma}(\frac 12,2),$ but is there a simple formula for the above (perhaps in terms of the polygamma ...
1
vote
1answer
38 views

Compute the conjunction probability of $contains$ operators in string

I have an unknown string S of length n. I have to given small string A1 and A2. Supposed ...
2
votes
1answer
78 views

Probability Statistics Question

I have this formula for determining $x$ and $y$'s effect on $$a\mapsto\frac{(xy/z)}{(xy/z)+ (1-x)(1-y)/(1-z)}$$ If this formula assumes x and y have equal affect on a (say 50% each), how would i ...
1
vote
1answer
94 views

Mean and variance for the raincoat problem

Suppose $n$ people each have a hat which they check at a hat room. Suppose that hats are returned randomly. Let $Y$ be the number of people who get their own hats. Find $E(Y)$ and $Var(Y)$. After few ...
1
vote
2answers
83 views

Maximizing discrete probability

I'm stuck with the following problem: Let's assume we have two buckets: bucket one contains $k$ white spheres and $l$ red spheres. Bucket two contains $n-k$ white spheres and $n-l$ red spheres (n a ...
0
votes
2answers
48 views

If $X,Y$ ~$U(0,1)$ what is the distribution of $Z=0.5x^{2}+0.5y^{2}$?

I have some trouble with it.. the question is: $X,Y$ uniformly distributed $U(0,1)$ than $\frac{1}{2}(x^2+y^2) $~$exp(1)$... I am not even sure it is correct.. I know that if $X,Y$~$N(0,1)$ than it is ...
1
vote
1answer
55 views

ALternate solution to a probability problem

Here is the problem: $A$ and $B$ roll a dice taking turns with $A$ starting this process. Whichever one rolls the first $6$, wins. Find the probability of $A$ winning. I know how to solve this ...
0
votes
1answer
42 views

Show that $\Pr(S_N\in A\mid N=n)=\Pr(S_n\in A)$

Let $X_1,.\ldots,X_n$ be i.i.d. random variables and $N$ be a positive integer-valued random variable, which is independent from the sequence. If $S_n=\displaystyle\sum\limits_{i=1}^{n} X_i$, then ...
1
vote
0answers
52 views

Spectral Representation for a real valued process

So I just finished reading a section in a book which discusses how every stationary stochastic process $\xi(t)$ can be expressed as $\xi(t)=\int_{\mathbb{R}}e^{it\lambda}\,dZ(\lambda)$ where ...
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1answer
42 views

Expected Value and coding

last week I had a test in which I didn't completely understand a question the teacher made: he asked to calculate the Entropy of a 3-extended source of two values (0 and 1) that have 1/3 and 2/3 ...
1
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1answer
103 views

what is wrong in my answer and what will be the correct solution for this probability question ?

A and B play a game where each is asked to select a number from 1 to 5. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial is - ...
3
votes
2answers
433 views

Russian roulette should a player pull the trigger or spin the cylinder

Two men plays Russian roulette. In revolver there are 2 bullets in consecutive chambers( 2 bullets are in 2 chambers next to each other). One man spun the cylinder, pulled the trigger and he is fine, ...
1
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1answer
42 views

Homework help with Standard Normal Distribution

I have a homework problem in which I'm not certain where to start: Let $X$ be a random variable with $N (0, 1)$ distribution. Show that $E(X^n) =\left\lbrace{\begin{array}{cc} 0 & \text{if $n$ ...
1
vote
1answer
36 views

Convergence to a finite random variable

The Martingale Convergence Theorem is typically stated that we have convergence to a (finite) random variable $X$ as $X_n \to X$ a.s when the conditions of the theorem are satisfied. What is meant ...
0
votes
1answer
96 views

Find total size/cost of list of IP addresses which is created randomly

I am creating a random IP addresses around 1 MILLION with the help of random function. I want to know what total cost of my structure is or how big is my data structure? Repetition of prefix than NO ...
1
vote
1answer
73 views

when it is conditional entropy minimized?

for example let us consider following table know that entropy of variable is maximum when it is equally distributed,all of it's variable has equal probability,but what about joint entropy or ...
0
votes
0answers
59 views

Proving properties of Random Graphs

I am asking the question on a slightly abstract level and it may depend on the specifics but it would be great to have related references or ideas. Consider the random graph model $G_{n,p}$ where its ...
0
votes
1answer
63 views

Compute a conditional probability of normal random variable

Suppose $X, T$ are continuous random variables, and $X \sim \mathcal{N}(0, 1)$, $T$ have density function $f_T$. (But $X,T$ do not have joint density) Is there any way to compute the following ...
1
vote
3answers
53 views

Covariance Issue with a joint distribution

Here is the problem: Let $X$ and $Y$ be continuous random variables with joint density function: $$f(x,y) = \frac{8}{3}xy\text{ for }0 \le x \le 1\text{ and }x \le y \le 2x $$ I'm trying to find ...
1
vote
1answer
39 views

Intuition of Markov structure and how variables relate in the structure

Assume three r.v. $X_1, X_2, X_3$. They are conditionally independent in the following way: $ X_1 \perp X_3 | X_2$ We have that: $$P(X_3 | X_2) = P(X_3| X_2, X_1)$$ In the notes I am reading it ...
4
votes
1answer
151 views

Why does this determinant have a continuous density at zero?

This question is a simplification of my previous question. I think this is easy, but I don't have a strong enough background in probability. Let $A$ be a random $n\times n$ real matrix that satisfies ...
0
votes
1answer
46 views

Probability of a dice being in a set of dice

I have no formal background in math, statistics, or anything. Just trying to figure out a fun problem with a game of dice. Lets say you have 3 people sitting a table rolling dice (including ...
0
votes
0answers
29 views

Simple question about conditional expectation

Let $X, Y,$ and $Z$ be random variables. Assume $X$ is independent of both $Y$ and $Z$. I know that $E[X|Z]=E[X]$. But is it true that $E[XY|Z]=E[X]*E[Y|Z]$? Sorry for not using formatting.
1
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3answers
97 views

how to write a joint distribution when using one random variable as a selector

Suppose there are 3 coins, A, B, and C, toss A first. If it is a head, toss B, otherwise toss C. Can I write this as a joint distribution of 3 random variables? Is this correct? $$ ...
0
votes
1answer
20 views

Vantage point tree question

I'm stuck in understanding the 1993 vantage point tree paper: http://aidblab.cse.iitm.ac.in/cs625/vptree.pdf It defines some things first: So if $x\in[0,1]$, then $P(x)$ is the probability of the ...
3
votes
1answer
52 views

Interesting Problem - Computing CDF

A rv X is an exponential distribution with parameter 1 and Y is a uniform distribution between 0 and 1. X and Y are independent. Define Z = min {X, Y}. Compute the CDF of Z ? I really have no idea ...
4
votes
2answers
1k views

Conditional distribution in Brownian motion

I need to prove the following: Let $X$ be a Brownian motion with drift $\mu$ and volatility $\sigma$. Pick three time points $s < u < t$. Then, the conditional distribution of $X_u$ given ...
1
vote
4answers
54 views

A simple conditional probability problem

Assume that two fair dice are rolled one at a time. Given that the sum of the two numbers that occured was at least $7$, compute the probability that it was equal to $7$. I tried computing the ...
0
votes
1answer
306 views

calculate channel capacity and maximum conditional entropy

i want to know when it is equal channel capacity or $I(X,Y)$ maximum or where $I(X,Y)=H(X)-H(X\mid Y)=H(Y)-H(Y\mid X)$ now if we have two random variable with some specific distribution ...
0
votes
2answers
54 views

X and Y are independent random variables and their distributions are..

X and Y are independent random variables and their distributions are.. $P(X=1) = 0.1 $ $P(X=2) = 0.2$ $P(X=3) = 0.3 $ $P(X=4) = 0.4 $ $P(Y=4) = 0.4 $ $P(Y=2) = 0.3$ $P(Y=3) = 0.2 $ $P(Y=4) = 0.1$ I ...
0
votes
1answer
21 views

find marginal density of $X$ where $X,Y$ have joint density $f(x,y)=c\cdot \exp (-(2x+3y))$ over the region $x>0$ and $x<y$.

Find marginal density of $X$ where $X,Y$ have joint density $f(x,y)=c\cdot \exp (-(2x+3y))$ over the region $x>0$ and $x<y$. I've found that $c=15$ for the joint density to be normalized. Then ...
0
votes
1answer
62 views

solving Venn diagram

A record survey was carried out among $70$ teenagers. The choice was out of three records: $A,G,P$ the result showed that $52$ teenagers liked $A$, $29$ liked $G$, and $37$ liked $P$. It was also ...
1
vote
1answer
53 views

Poisson probability for x greater than

When we calculate poisson probabilities for say $x> 4$, we usually find out by summing up probabilities of $x=0$, $x=1$, $x=2$, $x=3$ and $x =4$ and subtracting the aggregate from 1. But ...
5
votes
1answer
81 views

Drawing previously undrawn cards from a deck

Suppose you have a deck of $y$ cards. First, randomly select $y-x$ distinct cards and sign the face of each, then shuffle all the cards back in to the deck. Proceed as follows: Draw a card. If it is ...
14
votes
1answer
303 views

Can we qualitatively predict the strategy of the German and US teams in today's World Cup soccer match?

In today's World Cup soccer match between Germany and the US, both teams only need a draw to advance to the next round. There's been speculation about possible collusion, especially given the friendly ...
0
votes
0answers
39 views

One double integral elated problem

The bit I am stuck is the limits in the double integral. I tried X from 0 to uy and Y from 0 to infinity, this is obviously incorrect. I just want to know the complete double integral in the order ...
1
vote
1answer
36 views

Stochastic convex (conave) functions vs. convex (concave) function

Can someone help me understand the difference beween stochastic convex (conave) functions and convex (concave) function
3
votes
0answers
1k views

Problem with the expectation of a maximum of independent gamma distributed random variables

Having a problem with the expectation of the maximum among $n$ independent random variables $ X_1, X_2 \dots X_n$ all ~ the same class of distributions but not necessarily the same mean and other ...
1
vote
1answer
35 views

Calculation of conditional variance

I'd like to ask, whether I did this task correctly. We have two r.v. $X,Y$. Firstly, $Y$ has Bernoulli distribution $\mathbb{P} \left( Y=1\right)=a \; \; \mathbb{P} \left( Y=2\right)=1-a$ Moreover ...