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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33 views

Comparing odds of winning

Started to play the national lottery and noticed I can play combinations of numbers instead of just plain 6 out of 49. Since my math knowledge is fairly basic I am wondering which has the better ...
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63 views

Is this a beta distribution?

I'm currently implementing something from a paper and at one point it describes using a beta distribution of the form: $Prob(p) = (n + 1)Bi(n, p; m)$ where $Bi$ is the binomial distribution, $n$ and ...
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67 views

An irreducible Markov chain is a martingale

Let $\{X_n\}$ be an irreducible Markov chain. Does exist example of such $\{X_n\}$ which is also a martingale given that: a. $\{X_n\}$ is recurrent with finite number of states (but bigger ...
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1answer
37 views

Expected value with probabilities

Joel owns a lawn care business and recently performed some research on the size of 50 lawns that he takes care of. Joel recalls that he is expected to take care of a total of $21$ acres of lawn for ...
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2answers
75 views

Trouble with samples in a normal distribution

I'm okay with solving regular normal distribution questions (where X is a normal random variable with mean $\mu$ and standard deviation $\sigma$). However, we're currently dealing with samples within ...
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1answer
95 views

Integral of a Gaussian process has Gaussian Distribution

(1) How can we prove that the integral i.e. $\int_{a}^{b} X(t) dt$ (or any linear functional) of a Gaussian process $X(t)$ has Gaussian distribution? (2) And how can we find that distribution in the ...
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48 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
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0answers
42 views

If $E(|X|\log|X|)<\infty$ then is $E\left[\frac{|S_n|}{n}\ \log\left(\frac{|S_n|}{n}\right)\right]<\infty$?

I am trying to finish a homework problem in my probability class. I think I am at the end of my problem if I can show that $$E(|X|\log|X|)<\infty$$ implies that $$E\left[\frac{|S_n|}{n}\ ...
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2answers
80 views

Verify the joint probability function

I had a question I was hoping for some help on: There are 8 similar chips in a bowl: 3 marked (0;0), 2 marked (1;0), 2 marked (0;1), and 1 marked (1;1). A player selects a chip at random and is given ...
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1answer
133 views

Show that $F(x,y)=1$ for $x+y\ge 0$ and $F(x,y)=0$ otherwise does not define a joint CDF

Let $F(x,y)=1$ for $x+y\geq 0$ and be zero otherwise. Show that $F$ cannot possibly be the joint distribution function of a pair of random variables. Ok so basically I need to show that there can't ...
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2answers
202 views

Show that $\prod (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$

Let $A_n$ be independent events with $P(A_n) \neq 1$. Show that $\prod_{n=1}^{\infty} (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$ It kind of looks obvious but I really have no idea how to prove it. Can ...
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1answer
267 views

Compute $E(4XY/(X^{2} + Y^{2} + 1))$ for $X$ and $Y$ independent and uniform on $\left[0,1\right]$

Calculate $$ E\left(4XY \over X^{2} + Y^{2} + 1\right). $$ Right now I have gotten to $$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{4xy \over x^{2} + y^{2} + 1}\,{\rm d}y\,{\rm d}x. $$
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2answers
434 views

Find three events that are dependent but pairwise independent

Let $(\Omega, \mathcal F, P)$ denote the probability triple for the discrete uniform distribution on the set $\{1,2,3,4\}$. Q. Give an example of three dependent events with probabilities strictly ...
3
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1answer
64 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 ...
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0answers
21 views

Proving the sample variance has a chi squared critical value

Let $X_1, . . . , X_n$ be independent normal observations with means $µ = 0$ and variances $σ^2$. For testing the null hypothesis $H_0 : σ^2 = 1$ versus the alternative $H_a : σ^2 > 1$ show that ...
2
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1answer
30 views

Independent events?

Say we toss a coin that is fair twice we have sample space $:[HH,TH,HT,TT]$. Let's say $A$ is the event with the first throw a head. $A=[HH,HT]$. $B$ is the event with the second throw is a head. ...
2
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1answer
22 views

Question about probabilities of independent events?

Consider a fair coin toss. Let $H$ be the event that a heads was thrown. Let $T$ be the event a tail was thrown. Given $H$ and $T$ are disjoint are then independent? I don't really know how to do ...
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0answers
55 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 ...
2
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1answer
72 views

Show that if the sum of an diverges, no discrete probability space can contain independent events

Suppose that $0\leq p_n\leq 1$, and put $a_n= \min \{p_n, 1-p_n\}$. Show that if $\sum a_n$ diverges, then no discrete probability space can contain independent events $A_1, A_2, \ldots$ such that ...
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1answer
57 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 $a(X_1 - X_2 + 1)^2$ has a chi-squared distribution. ...
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2answers
92 views

probability of rolling same number from 3 dice

How can I calculate the probability of getting the same number from rolling 3 8 sided dice? I know there are similar questions but I have been out of study for a long time and I need to get a firm ...
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0answers
19 views

Calculate $P(X_{16}=2|X_0=0)$

Given a Markov Chain with three states 0,1,2 with the following State Transition Probabilites: $$M = \left( \begin{array}{ccc} 0.3 & 0.3 & 0.4 \\ 0.2 & 0.7 & 0.1 \\ 0.2 & 0.3 ...
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1answer
142 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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2answers
216 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 ...
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1answer
41 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 ...
3
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2answers
171 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. ...
2
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1answer
55 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 ...
3
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2answers
52 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 ...
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1answer
599 views

Probability in Rolling dice 6 times

If I have one die and I'm rolling the dice $6$ times. What is the probability that in the all $6$ times the result will be the same? I know that the probability for each number in $6$ sides dice is ...
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2answers
9k views

Probability of rolling the same number twice

Math novice here. With a 10-sided die, the probably of rolling '1' is 10%. I'm tempted to think the probability of rolling '1' with two consecutive rolls is 20%. Would I be correct? Not sure if I ...
2
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1answer
627 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 ...
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2answers
59 views

If X is a random variable how do we show that $E(|X|)=0 \iff P(X=0)=1$

If X is a random variable how do we show that $E(|X|)=0 \: \iff \: P(X=0)=1$ I see that $-|X|\le X \le |X|$ and so $|E(X)| \le 0$ and thus $E(X)=0$ but how do I show that this implies $P(X=0)=1$.
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3answers
5k views

Probability of passing this multiple choice exam [duplicate]

A multiple choice exam has 175 questions. Each question has 4 possible answers. Only 1 answer out of the 4 possible answers is correct. The pass rate for the exam is 70% (123 questions must be ...
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1answer
43 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 ...
2
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1answer
139 views

How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a ...
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0answers
31 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 ...
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1answer
38 views

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

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

$E_n =\lbrace X_n > X_m \ \forall m < n \rbrace $ are independent

I'm stuck with this exercise. Suppose $(X_n)$ are independent random variables defined on $(\Omega, \mathfrak{F}, P)$ with the same p.d.f. Let $E_1 = \Omega$ and for $n \geq 2$ $$E_n =\lbrace X_n ...
2
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1answer
37 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 ...
2
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1answer
163 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 ...
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2answers
72 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 ...
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1answer
55 views

Explicit CDF associated to Gamma PDF

Let the distribution function of $X$ for $x>0$ be: $$F(x) = 1 - \sum_0^3 \frac{x^ke^{-x}}{k!}$$ what is the density function of $X$ for $x > 0$? This is what I'm thinking: $$ ...
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0answers
47 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. ...
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1answer
52 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 ...
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3answers
134 views

Probability: Draw your own card

In the anime HunterXHunter, there is an episode where 24 people are assigned a numbered ID tag, and each person draws a card with one of those numbers on it. Each person is supposed to hunt the person ...
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3answers
40 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 ...
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1answer
50 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 ...
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3answers
60 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.
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3answers
53 views

How can I calculte the probability of $X$ with a Generlized Hyperbolic Distribution?

I would like to know how to calculate the probability of $X$ when I have fitted a Generalized Hyperbolic Distribution to my data set. The depth of my knowledge is basic t-tests and z-tests. I am ...