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
12 views

Does this argument suffice to show a “record” occurs at time n with probability 1/n?

I think it does, but, in addition to checking for correctness, I'd like to know what other argument we might use. Let $X_1, X_2,...X_n$ be be a sequence of independent identically distributed ...
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0answers
14 views

Prove about weak convergence

Let $X_n\Rightarrow X$ and $Y_n\Rightarrow c$. Show what $X_n+Y_n\Rightarrow X+c$. Prove: There exists sequences of random variables $(X^{(*)}_n)$ and $(Y^{(*)}_n)$ such that $(X^{(*)}_n)$ and ...
-1
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0answers
16 views

Bayes Theorem and Probabilities [on hold]

Suppose that economic outcomes can be classified as either good or bad. Governments differ in ability and this affects the likelihood of good outcomes. There are two types of governments: high ability ...
0
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0answers
17 views

Bipartite graph with $2 \times 10^{6}$ vertices, I need help with removing edges from the graph.

Let G be a bipartite graph. The number of vertices are equal to $2 \times 10^{6}$. Every node is of degree 10. We remove every edge with Probability $2^{-0,1}$. Show that the number of nodes after ...
2
votes
1answer
14 views

Fatou for weak convergence

I want to do exercise 3.2.4 from Rick Durett, Probability: Theory and Examples page 86. $$\text{Let } g\geq0 \text{ be continuous. If }X_n \Rightarrow X_{\infty} \text{ then } \liminf_{n\rightarrow ...
0
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0answers
10 views

Mean for seat allocation

There are a set of kids (let's say 30) asked to sit in a row of seats, leaving at least one empty seat between them until all seats are filled. At the end, how do I calculate mean of the fraction of ...
2
votes
2answers
14 views

Variance and expected values by dices, how does addition work?

I have read through some stuff and I am confused now. If we have a fair die and we just roll once, the expected value is going to be 3,5 and the variance is 2,916. Well, it is easy to count by one ...
0
votes
1answer
23 views

Question about sums of normal random variables

I have independent random variables $X_1$, $X_2$ such that $X_1 \sim N(1,1)$ and $X_2 \sim N(2,2)$, and I'm trying to find a constant $a$ such that $c(X_1 - X_2 + 1)^2$ has a chi-squared distribution. ...
1
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2answers
29 views

Probability involving chess board

if 2 cells are chosen at random on a chess board what is the probability that they will have a common side i tried solving the question by considering different cases for the cells on: 1. corner 2. ...
0
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0answers
31 views

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, ...
0
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1answer
25 views

Total Probability theorem and Bayes theorem

Two reinforced concrete buildings A and B are located in a seismic region. It is estimated that an impending earthquake in the region might be strong (S), moderate (M), or weak (W) with probabilities. ...
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0answers
13 views

Find all random variables whose distribution satisfies an equation

The problem I have to solve is formulated as follows: Find all random variables such that if $Y$ has the distribution $N(0,1)$ and $X, Y$ are independent then $X+Y$ has the same distribution as ...
0
votes
1answer
19 views

Conditional probability density function

Let $\theta$ be the parameter of the probability density function $f(x)$. If it is mentioned that $f(x|\theta)$ be the conditional probability density function, then what does $f(x|\theta)$ mean? ...
1
vote
1answer
24 views

Writing random variable formulas with set notations, What is the problem?

Is it wrong to write $\displaystyle P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$ when $X$ and $Y$ are random variables? As I know a random variable is a function and therefore has a range and the two ...
1
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0answers
19 views

Estimate probablity: Chernoff bound

Two players $A$ and $B$ are playing following game: They throw cube. When thrown number $k$ and $k$ is even then player $A$ get $k$ points. When thrown number $k$ and $k$ is divisible by $3$ then ...
0
votes
1answer
27 views

What does it mean $\int_1^\infty\frac{F(y)}{y^2}\mathrm dy$?

Which type of functions will satisfy this? $$F: [1,\infty) \to [0,\infty)$$ $$\int_1^\infty \frac{F(y)}{y^2} dy \leq 1$$
3
votes
1answer
19 views

Prove that if $E(X\log X)<\infty$ then $E(\sup_n |S_n|/n)<\infty$.

This is part 2 of a two part question. In the first part, we were asked to show that if you had a non-negative sub martingale $M_n$ then $$\sup_n E(\sup_{k\leq n} M_k)\leq \sup_n 2E(M_n \log M_n)+2$$ ...
2
votes
2answers
45 views

Find $E(|X-Y|^a)$ where $X$ and $Y$ are independent uniform on $(0,1)$

Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a>0$. My working: Define $W=1$ if $X>Y$ and $W=0$ if $X<Y$. We seek ...
2
votes
1answer
15 views

Conditional expectation of $Y_1$ given that $\sup Y_i=z$, for $(Y_i)$ i.i.d. uniform on $[0,\theta]$

Suppose that $Y_1,\ldots,Y_n$ are random variables independently and identically distributed as uniform on $[0,\theta]$ for some $\theta>0$. How do I find the conditional density of $Y_1$ given ...
0
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2answers
38 views

What is the logic behind the probability of getting 'four of a kind' in poker?

This hand ($5$ cards of $52$) has the pattern $AAAAB$ where $A$ and $B$ are from distinct kinds. The number of such hands is $\binom{13}{1} \binom{4}{4} \binom{12}{1} \binom{4}{1}$. The probability ...
1
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2answers
60 views

How can I reword this problem illustrating a scenario that needs Bayes Theorem to solve?

Taken from Stat Trek, an example explaining Bayes Theorm http://stattrek.com/probability/bayes-theorem.aspx Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent ...
1
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1answer
16 views

Help finding a probability density function

I am having a bit of trouble with this: Let $U=Z^2$ where Z is the probability density function of the standard normal distribution. So, $f_z(z) = \frac{1}{2\pi} e^{\frac{-z^2}{2}}$ I want to use ...
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0answers
17 views

Estimating probablity with using Chernoff inquality

I would like to estimate that probablity that $X_k>1250$. So, $X_k=Y_1 +...Y_{10^6}$ $Y_i = 1 $ with prob = $\frac{1}{10^3}$ $EX_k = 1000 $ Chernoff that I will use: $$P(X_k \ge (1+\epsilon)EX_k) ...
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0answers
30 views

Is this an easy conditional probability question?

Fifty-two percent of the students at a certain college are females. Five percent of the students are majoring in computer science. Two percent of the students are females majoring in computer science. ...
0
votes
1answer
24 views

Expected Value Coins Question

If I were to flip n coins and compute the product of the number of heads versus the number of tails what would be the expected value of this product? My logic: In n coin flips n/2 coins will be ...
0
votes
3answers
22 views

Chances of random number belong to a given set

I have 23 elements and 7 of them belong to a given set. 5 of these 23 elements will be picked randomly, I want to know the chances of at least one of those selected 5 elements belong to the ...
0
votes
1answer
24 views

Flipping an unfair coin n times

I’m flipping an unfair coin $n$ times. $\mathbb{P}[X=head]=p$ where $p \neq \frac{1}{2}$. What is the probability “head” appears an even number of times? Thank you in advance for your time an ...
0
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0answers
22 views

Repeated coin flips probability [on hold]

Assume in an experiment, one flips a coin $L$ times. This experiment is repeated T times. Assume the $k$'th flip for all possible $k$ values ($1 \le k \le L$) among all experiements. If the head ...
0
votes
3answers
39 views

Probability of drawing n distinct values out of {1,…,n^3}

I draw uniformly at random $n$ values out of $\{1,...,n^3\}$. I want to lowerbound the probability of getting $n$ distinct values.
2
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0answers
32 views

Can anything be learned about a probability distribution *directly* from its characteristic function?

Some preliminaries: I know that one can take the inverse Fourier transform to get back the pdf...that is not what I am after. My question is whether the characteristic function, qua function, tells us ...
0
votes
1answer
22 views

Computing Average Number of Successes When Randomness is Involved

I am attempting to write a program that will compute the average amount of a particular product produced when randomness is involved. Let's say that I am trying to produce some widget. Whenever the ...
0
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0answers
19 views

Calculating Power of a Paired T Test

$ 239$ subjects had their cholesterol measured, and then were put on high-fiber diets. After a month on the high-fiber diet, the cholesterol was measured again. The mean LDL cholesterol level before ...
0
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0answers
11 views

How can I scale the covariance matrix which represent a gaussian distribution ? [on hold]

I have a model genrated by using GMM the output is the mean and covariance matrix .I need to scale the cov matrix .for example I want to double the elipse that represent this gaussian .
-1
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1answer
27 views

Probability related question, Permutations, combinations [on hold]

Im doing a practice problem for an upcoming test, I had a hard time figuring out this question, could anyone walk me through it?
2
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0answers
87 views

Is there a real problem to which $1$ radian is the answer?

I can't recall if I've ever seen any problem related to angles, in math or engineering books, that would result in an answer like $$\alpha=1 \ \ \text{radian}.$$ The answers to such questions, I ...
1
vote
1answer
42 views

What does it mean that an expected value does not exist?

$X$ is a random variable with pdf $f$ and $g: \mathbb R \to \mathbb R$ is a measurable function. Before I start operating with $E[g(X)]$ I need to show that it exists. What does it take to show it? ...
0
votes
1answer
11 views

What is the probability that $x$ will not work due to failure rate $0.0111$

I've tried using the probability mass function for binomial distribution in this case but it seems to not be the appropriate approach unless I calculated wrong. How am I supposed to approach this ...
2
votes
1answer
53 views

Expected number of red balls removed from an urn before the first black ball

Question: An urn contains n+m balls of which n are red and m are black. They are withdrawn from the urn one at a time and without replacement. Let $X$ be the number of red balls removed before the ...
-1
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0answers
23 views

Cube of Brownian motion [on hold]

Find all $H_t$ so that: $B^3_T = \int_0 ^T H_t dB_t$ $\int_0^T B^3_tdB_t = \int_0^T H_t dB_t$ where $B_t$ is a Brownian motion.
3
votes
1answer
41 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
3
votes
1answer
22 views

Using Conditional Jensen inequality proof the following

$X_1,X_2,\ldots,X_n$ are i.i.d. random variables, $X_1>0$, $E[X_1]=\mu$, $E[X_1^k]<\infty$ for $1<k \leq2$. Proof: $$ E\left[\left(\frac{1}{n}\sum_{i=1}^nX_i\right)^k\right]\leq ...
-2
votes
0answers
27 views

conditional probability proof 3 varables [on hold]

Suppose that $\mathcal a$ ,$\mathcal b$ and $\mathcal c$ are dependent variables. $$\mathbb P(a \mid b) = \sum \mathbb P(a \mid b,c) \ \mathbb P(c \mid b)$$ can anyone explain it how we get it?
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votes
2answers
25 views

Rolling dice probability by solving inequlity

I was trying to solve a problem where I have to find the probability of the sum of $\mathcal 3$ rolls of a die being less than or equal to $\mathcal 9$. In order to solve the problem I try first to ...
0
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0answers
23 views

Probability of a sequence of urn draws having some pair of draws with a minium number of “matches”?

I have $U$ urns. Each urn contains some sequentially numbered balls (not necessarily the same count between urns) $1, 2, 3,... N_u$. I draw one ball from each urn $1, 2, 3,...U$ in turn, and note ...
0
votes
0answers
21 views

How to use joint probability density to check for independent events?

Suppose that the joint PDF of $X$ and $Y$ is as follows: $$ f(x) = \begin{cases} 24xy & \text {$x \geq 0, y \geq 0, x+y \leq 1$}\\ 0 & \text {otherwise ...
1
vote
1answer
17 views

Covariance of $2$ variables

I am given two random variables $X$ and $Y$. I am also given that $\mathbb{E}(Y|X)=\mathbb{E}(Y)=\mu_y$ and $\mathbb{E}(X)=\mu_x$. So if I need to calculate the covariance of $X$ and $Y$, ...
5
votes
1answer
34 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
1
vote
1answer
24 views

Question about asymmetry of chi-square distribution

Let $X_1,\dots,X_n$ be a set of i.i.d. chi-square random variables with $k$ degrees of freedom. Consider the statistic $\arg\max_i\{|X_i/k - 1|\}=X_{\alpha}$. I wonder about the probability that ...
0
votes
2answers
16 views

Multinomial Coefficients Dice Problem

If 7 balanced dice, are rolled, what is the probability that each of the 6 different numbers will appear at least once? My attempt: $p=\frac{7!}{2!6^6}$ So if 6 different numbers need to appear, ...
2
votes
3answers
34 views

Probability of balls in boxes

If $12$ balls are thrown at random into $20$ boxes, what is the probability that no box will receive more than $1$ ball? So my book says the answer is: $\displaystyle \frac{20!}{8!20^{12}}$ However ...