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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0
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1answer
45 views

Why do we need to use random variables

In my statistics textbook (The Practice of Statistics by Starnes, Yates, and Moore) an example is given. In it, 21 students are each given three glasses of water. Two are filled with tap water and one ...
2
votes
2answers
43 views

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it's the same or am I ...
1
vote
1answer
42 views

P.d.f of $X_{(1)}/X_{(n)}$

Let $X_{i} \sim U(0, \theta) $ and $X=(X_1,\dots,X_n)$. Find the pdf of $$ \frac{X_{(1)}}{X_{(n)}}$$ I coulxnt find a way of doing it that looks convenient. Any idea? P.s: $X_{(i)} $ are the order ...
0
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0answers
35 views

Question on Markov chain [closed]

I came across this problem while reading about Markov chain.... N students enter a clean room facility to do experiments. They have to leave their shoes outside the lab. After finishing the ...
3
votes
0answers
26 views

Itô Excursion Measure

I am looking for any source of information regarding Itô Excursion Measure (for Brownian Motion). I am looking for a selfcointained reference (Though I have basic knowledge on Local Times and Poisson ...
1
vote
2answers
42 views

Maximum of Three Uniform Random Variables

Here is the question, I am studying for exam P, and am using a study guide with a solution guide. I am stumped on this problem, and the solution in the back was very confusing. Any clarity I can get ...
0
votes
2answers
57 views

Probability distribution of $\min(X,Y)$ given that $\max(X,Y)>1/2$ [closed]

Suppose $X$ and $Y$ are two independent random variables. What is the value of $\Pr[\min(X,Y) \leq z \mid \max(X,Y) >1/2]$? They both follow a Uniform distribution with parameters 0 and 1
2
votes
0answers
37 views

What is the probability that the dj will play only R&B and Jazz songs alternatively?

I just want to check if my answer is correct. Every Monday morning from 0700-0800 hours, the radio disk jockey (dj) on duty must play 8 songs on radio; the songs are pre-selected by his manager and ...
0
votes
0answers
12 views

Distribution of time derivative of a random variable

I am wondering how can i find the distribution of the derivative of a random variable. As a simple case i would like to start with Normal distribution, but would like to understand this for ...
1
vote
1answer
36 views

Distribution of $|X|$

Find the distribution of $|X|$ if $X \sim N(\mu, 1)$ My attempt: $$P(|X| \leq x) = F_X(x) - F_X(-x)$$ If $F$ denote the cumulative distribution of $X$, then $$P(|X| = x) = ...
0
votes
1answer
18 views

Show $P(X=n)=\left(\frac{1}{2}\right)^{n+1}$ for Poisson variable with exponentially distributed $\lambda$

I'm supposed to do the following, any help/pointer is appreciated: Suppose $X$ is Poisson distributed with mean $\lambda$. Suppose $\lambda$ is exponentially distributed with mean $1$. Show that ...
1
vote
2answers
25 views

What is the probability that the dj will play 3 songs in any order that is not consecutive?

Every Monday morning from 0700-0800 hours, the radio disk jockey (dj) on duty must play 8 songs on radio; the songs are pre-selected by his manager and the dj may play any eight of them randomly. ...
0
votes
1answer
59 views

Derivation of Black-Scholes equation by riskless portfolio

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
0
votes
1answer
23 views

What is the probability that the owner is on the review committee and there is at least one member from each of the other three staff categories?

A company has twenty two staff members: the owner, three senior executives, fourteen junior executives and four support staff. A salary review committee consisting of eight members of staff needs to ...
0
votes
0answers
26 views

Bayes Theorem with multiple observations

Let $H \in \{1,..,K\}$ be a discrete random variable and $e_1, e_2$ be observed values of 2 other random variable $E_1$ and $E_2$. We wish to calculate the vector ...
1
vote
1answer
11 views

What is the probability that there are strictly fewer support staff on the committee than senior executives?

A company has twenty two staff members: the owner, three senior executives, fourteen junior executives and four support staff. A salary review committee consisting of eight members of staff needs ...
0
votes
0answers
8 views

Given MTTF and Number of Items, how to calculate failing parts with Time?

I am wondering how to make use of MTTF Here is the situation, I am given an MTTF for an item type x and a certain demand for that item in the next 25 years, say 100 parts that will be in operation ...
0
votes
2answers
21 views

Strongest 'average' for a diverse set of numbers?

I have a set of numbers consisting of two general size numbers: size 'a', and size 'b' which are about three times bigger in size than size 'a'. There is some variation and the list might look like ...
3
votes
1answer
24 views

If $\sum_{n \geq 1}X_n$ converges a.s. then $\forall a > 0: \sum P(|X_n|>a) < \infty$

I'd like to show that for $(X_n)_{n\geq 1}$ a sequence of real-valued and independent random variables, If $\sum_{n \geq 1}X_n$ converges a.s., then $\forall a > 0: \sum P(|X_n|>a) < ...
0
votes
1answer
33 views

How to increase winning chance in lottery [on hold]

Let us imagine such kind of lottery game :lottery machine is running and randomly is selecting $7$ number from $1$ to $36$(including).out of this $7$ numbers,$6$ are basic or in other word ,jackpot ...
0
votes
4answers
65 views

Expected value of $e^X$ where X is geometrically distributed

Random variable $X$ is geometrically distributed with parameter $2/3$, that is $\mathbb{P}(X=k)=2/3^k$, $k=1,2,...$ I have to find $\mathbb{E}e^X$. So let $Y=e^X$. Then ...
3
votes
4answers
2k views

If a fair die is thrown three times, what is the probability that the sum of the faces is 9?

If a fair die is thrown thrice, what is the probability that the sum of the faces is 9? I did like this. The total number of cases is $6^3=216$ Now,the number of solutions of the equation $x + ...
2
votes
2answers
59 views

If $ P(A) = 0 $ is $ A $ a null event?

I know that $ P(\text{null event}) = 0 $, but is the reverse true? i.e. if $ P(A) = 0 $ is $ A $ a null event? I'm not too sure I even understand what a null event is, to be honest. Could anyone give ...
0
votes
1answer
20 views

Let X be a random variable with PDF fx. Find the PDF of the random variable |X| in the following

Here's my question: X is uniformly distributed in the interval $[-1,2]$. Find pdf of $|X|$... So I did P($|X| \le x$) = P($-x \le X \le x$)... From here I'm not too sure how to proceed. I know the ...
0
votes
0answers
24 views

Relationship between quotient of sum of exponentials and uniform distributions

Let $X$, $Y$ and $Z$ be iid with $P(X>t)=e^{-t}$ for $t>0$. Let $U$, $V$ be independent uniform on $[0,1]$. Let $A=\min(U,V)$ and $B=\max(U,V)$. Show that $(A,B),$ and $(X/(X+Y+Z), ...
2
votes
5answers
75 views

Mean and variance of the sum

This problem showed up on my exam: A box contains 10 slips bearing the numbers $1,2,...,10$. We take out $2$ slips at random without replacement and add the two corresponding numbers. Find the mean ...
1
vote
1answer
27 views

Probability that $f(x,y,z)>0$ given the variables follow normal distribution

Assuming that variables $x,y$ and $z$ follow the Gaussian distribution with $\mu_x=\mu_y=\mu_z=1000000$ and $\sigma_x=\sigma_y=\sigma_z=200000$, what is the probability that $$f(x,y,z) = ...
1
vote
1answer
16 views

Probability of a single variable from a Moment Generating Function

This is from the A/S/M study guide, and the answer is listed, I just don't understand how he's arriving at the answer... I'm sure it's something simple I am missing! I have two identically ...
1
vote
5answers
76 views

What is the probability that no married couples are among the chosen?

Eight married couples are standing in a room. 4 people are randomly chosen. What is the probability that no married couples are among the chosen?
1
vote
1answer
42 views

What is the probability that they are married [closed]

Eight married couples are standing in a room. (a) If two people are chosen at random, what is the probability that (i) they are married; (Answer: 0.0167) I can't get their answer. Is it right? ...
1
vote
1answer
39 views

If $f$ is a pdf can we construct $g$ such that $x\sim U[0,1)$ implies $g(x)\sim f$

Let $f$ be some pdf over $[0,1)$. Here is my question: does there always exist an infinite partition $\{X_{s}\}_{s\,\in\, \mathrm{support}(f)}$ of $[0,1)$ such that if we define $g(x):[0,1)\rightarrow ...
1
vote
1answer
26 views

Can the distribution of $Z=kX+(1-k)Y$ conditional on $X\geq Y$ be decomposed into the distributions of $(X,Y)$ uniquely?

$X$ and $Y$ are two independent random variables that take values on interval $[0,1]$. Let $Z=kX+(1-k)Y$ where $0\leq k\leq 1$ is a known constant. Suppose the distribution of $Z$ conditional on ...
-4
votes
1answer
78 views

Will you be rich? [closed]

You have $\$1000$ and you start betting at $\$1$. The win rate is $50\%$. If you win you start betting at $\$1$ again, if you lose you bet double your previous bet. Will you be rich? Example
-6
votes
1answer
26 views

Probability of an actual event [closed]

What is the probability of all four golfers scoring a birdie on the same hole during the same round of golf. The average handicap of the four golfers is 12.
1
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0answers
38 views

Estimators and confidence interval

Can someone explain me how to solve the following exercise? I don't like to post this kind of question, but in this case I have a really bad theory material and I would greatly appreciate a concrete ...
1
vote
1answer
33 views

Distribution of number of Poisson arrivals in interval

$X_1$ and $X_2$ are both Poisson processes. $N$ is the number of arrivals of $X_1$ in between two subsequent arrivals of $X_2$. Derive the probability density $f_N(n)$ of $N$. I wanted to start from ...
0
votes
0answers
41 views

Why is the covariance matrix of i.i.d. Gaussian variables the identity matrix?

What will be the covariance matrix if observed samples $\{x[0],x[1],...x[N-1]\}$ are IID according to gaussian pdf; $$p(x[n];u)=\frac{1}{{2 }}e^{-{|x[n]-\mu|}}$$ I want to find best linear unbiased ...
0
votes
0answers
27 views

Markov chain with Poisson distribution

Let $X \in \mathbb{R}^+$ and $Y \in \mathbb{Z}^+$ be the Random Variables (RVs) where the condition PDF $f_{Y|X}(y|x)$ follows a Poisson distribution as $$ f_{Y|X}(y|x) = ...
0
votes
2answers
31 views

What is the probability that the 3rd best club will appear in the final stage?

16 clubs are divided into 4 groups with the winner of each group entering the final stage of the competition. What is the probability that the 3rd best club will appear in the final stage? All the ...
7
votes
0answers
100 views
+100

How to estimate $Pr[vr_i=ur_i]$ in the presence of rotations

Suppose we want to compute the probability that for two different random vectors (with elements that are $0$ or $1$), denoted by $v$ and $u$, multiplying them with the rotations of a random vector $r$ ...
-3
votes
1answer
31 views

What is the probability that at least 2 of the 4 end balls are the same colour? [closed]

There are 2 identical sets of balls Each of them has: 3 blue balls, 2 red balls and one white ball. Each of these sets is placed in a line. What is the probability that at least 2 of the 4 end ...
0
votes
1answer
24 views

Supply the transition matrix for these (possible) Markov chains

Reading Grimmet, Stirzaker: Probability and Random Processes, which unfortunately doesn't have solutions. Trying to make sure I understand Markov chains. A die is rolled repeatedly. Which of these ...
1
vote
1answer
32 views

Covariance of a function of random variables

I want to find the covariance $K_X(t,t')$ of the following signal $X(t)$: $X(t)=\sum\limits_{n=-\infty}^{+\infty} A_np(t-nT)$ where $ p(t) = \begin{cases} \ 1 & \text{if } 0<t\leq T/2 ...
-1
votes
1answer
28 views

In how many of the possible arrangements will both end balls be of the same colour? [closed]

6 blue balls, 4 red balls, and 2 white balls are placed in a straight line. In how many of the possible arrangements will both end balls be of the same colour?
0
votes
3answers
45 views

What should n be so that the probability is less than 0.5 [duplicate]

n represents the number of people. The probability is that none of these people have a birthday on the same day. Neglect people that are born on 29 February. What should n be so that the ...
3
votes
0answers
77 views
+50

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
0
votes
1answer
29 views

Proof of Double Expectation of a Conditional Expectation

There is a proof of $$ E(E(Y|x)) = E(Y) $$ $Proof:$ WLOG, suppose X and Y are two continuous random variables. Let $E(Y|x)=m(x) =\int_{-\infty}^{\infty} yf(y|x)\, dy$ Then $$ E(E(Y|x))=E(m(x))= ...
0
votes
2answers
42 views

Existence of density function for a sum of 2 Random Variables

Let's suppose that $Y$ is the normal distribution and that $X$ is another random variable whose density function may or may not exist. Does it follow that $Y+X$ has a density function? I am reading ...
0
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0answers
28 views

On achieving the maximal correlation

I am reading the famous paper of Renyi, entitled "On measures of dependence" (see here1). He redefined the maximal correlation in a very general form for both discrete and continuous random ...
2
votes
1answer
45 views

Where can I find a set of probability problems?

Is there a database of solved probability problems available? I am currently studying probability (and statistics) and, while I think I have a decent grasp of permutations, combinations, conditional ...