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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3
votes
2answers
166 views

Calculating integral $\int_{0}^{\infty}x^2 \frac{f'(x)^2}{f(x)}dx$

This is a follow up question for this question: How can I calculate or simplify the following integral $$\int_{0}^{\infty}x^2 \frac{f'(x)^2}{f(x)}dx$$ If I know f(x) is a probability density ...
0
votes
5answers
86 views

Most likely length of random lines inside a sphere.

Take a sphere in $\Bbb R^3$ with diameter $d$. Now secants are drawn randomly through the sphere. Consider tangents as secants here. Any secant is equally likely. Also this. $L$ is the length of ...
0
votes
0answers
30 views

Use $E[(X\leq Y)^2]$ to obtain lower and upper bounds of $E[XY]$ when $X$ and $Y$ are not statistically independent.

Use $E[(X\leq Y)^2]$ to obtain lower and upper bounds of $E[XY]$ when $X$ and $Y$ are not statistically independent. I don't understand what $(X\leq Y)^2$ supposed to mean. Is it a typo error?
0
votes
1answer
74 views

Is this a matrix notation of standard error?

What is the matrix notation of the standard error? A friend is referencing the standard error as: $$SE^2=(XX^T)^{-1}\sigma^2$$ $$\sigma^2 = \frac{1}{n-p}\sum\limits_{i=1}^n |\hat{y_i}-y_i|$$ ...
0
votes
1answer
37 views

Probability of an event whose chance of occurring doubles once after the first event in a sequence

To explain my problem I'll use the following example. There is a bag with 9 blue balls and one red ball. On the first event, a ball is taken out of the bag (90% chance it is blue and 10% chance it is ...
-1
votes
1answer
55 views

How do I find the probability given this situation:

I own a fleet consisting of X battle ships. Y of the ships in my fleet have malfunctioned and are launching missiles at friendly ships. Each day, so long as there is at least one malfunctioning ship, ...
0
votes
1answer
36 views

Book Recommendation Please! [Casella Berger] Statistical Inference

I would appreciate your 2 cent on book recommendation. I have basic exposure to probability theory back in college (e.g. calc, stats, probability undergrad level) but haven't dealt with them for a ...
1
vote
2answers
22 views

Very Basic Probability: Lottery Chances

If each ticket for a lottery has a 1 in 258,890,850 chance... what happens if you buy 10 tickets? Is it: A 10 in 258,890,850 chance? A 1 in 258,890,840 chance? A 1 in 25,889,085.0 chance? What ...
-1
votes
3answers
87 views

I flip a coin 100 times. How likely will I get at least 65 tails? [closed]

If I flip a coin $100$ times (assuming that the odds are $50/50$), how likely is it that it will land on tails at least $65$ times?
2
votes
2answers
58 views

Expected value of $x^t\Sigma x$ for multivariate normal distribution $N(0,\Sigma)$

For standard normal distribution, the expected value of $x^2$ is $1$. A natural question is that in the multivariate case, what is the expected value of $x^t\Sigma x$ for multivariate normal ...
2
votes
1answer
61 views

Uniqueness of the uniform spherical distribution

Suppose that $X,Y$ are random vectors on some (possibly different) probability spaces mapping to $\mathbb R^n$ for some $n\in\mathbb N$. Suppose furthermore that $\|X\|=r>0$ for all realizations ...
4
votes
1answer
64 views

Splitting Line Segments and Finding Expected Value

Consider a line segment which has a length of $2n-3$. It is split into $n$ segments at random. It is guaranteed that $n\ge 3$ and $n\in \mathbb{Z}$. These smaller lines are then used as the sides of a ...
4
votes
1answer
70 views

Convergence of the integral $\int_0^\infty f(x)\frac{xf'(x/(1-1/N))}{f(x/(1-1/N))}\ \mathsf dx$ as $N\to\infty$

How can calculate this integral $$\lim_{N \to \infty} \int_{x=0}^{\infty}f(x) \frac{x f'\left(\frac{x}{1-1/N}\right)}{f\left(\frac{x}{1-1/N}\right)} dx$$ where $f(x)$ is a probability density function?...
0
votes
3answers
28 views

Probability of card sequence of 1,2,3

If I have $3$ cards numbered $1$,$2$ and $3$ and pick them randomly replacing the picked card for the next pick. My Attempt: I understand the odds of picking each card is $\frac{1}{3}$. The odds of ...
-1
votes
2answers
41 views

An urn contains $6$ red marbles and $4$ black marbles [closed]

An urn contains $6$ red marbles and $4$ black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?
1
vote
2answers
34 views

How to interpret probability of a nonrepeating event?

I'm wondering about the meaning of ascribing probabilities to the outcomes of nonrepeating events. As a concrete example, here in the UK we're pollsters are currently predicting the result of the ...
0
votes
1answer
20 views

Probability of Jon living on my floor or being my neighbor

I am moving into an apartment building of 4 floors, with 10 apartments per floor. My apartment has 2 neighboring apartments. Jon lives in the same building (and does not live in the same apartment as ...
1
vote
1answer
23 views

How to calculate the probability that the distance between two points is less than some value?

If I have two points $A=(x_A, y_A)$ and $B=(x_B, y_B)$ generated uniformly ate random in 2D Euclidean space. Here, I assume $0­\le x_A, x_B, y_A, y_B \le 100$. I would like to calculate the ...
0
votes
0answers
15 views

Given probability generating function (PGF) for $X$, find PGF for $Y=2x+1$

PGF for $X$ is given by $G_{X}(t)=t^2e^{t-1}$, where $t\in\mathbb{R}$. Find PGF for $Y=2X+1$. I started it doing this: $G_Y(t)=\mathbb{E}t^{2X+1}=t\cdot\mathbb{E}t^{2X}=t\sum_{k=0}^{\infty} t^{2k}\...
0
votes
1answer
19 views

Random subseries of harmonic series expected to converge, but how often does it?

Inspired by a previous question which I can't seem to find, what if we have $$X = \sum_{k=1}^{\infty}\frac{1}{k}\cdot P\left(U(0,1)<\frac{1}{k}\right)$$ That is, each term of the series will be $\...
6
votes
0answers
50 views

What is the probability that they have no common prime factor?

I am seeking a simple way to solve the following problem. It is an easy problem, but I don't like the way I solve the problem. I listed two sets of numbers and counted one by one first and then find ...
1
vote
1answer
31 views

Probability that at least 2 edges of $\Gamma_{n,N}$ shall have a point in common

In the classic paper of Erdos,Renyi On the evolution of random graphs[page 7] ,it is argued that the probability that at least 2 edges of $\Gamma_{n,N}$ shall have a point in common is given by $1-\...
2
votes
2answers
75 views

Probability: Balls in baskets

I'm self learning and I stumbled upon the following exercise but I'm not sure if I solved it correct as I'm very new to this. Problem: 7 balls fall independently into 7 baskets. Let $X_i$ = number of ...
2
votes
2answers
60 views

Are real numbers generated uniformly at random guaranteed to be unique?

Suppose I can generate numbers uniformly at random from an infinite set, such as: $$r \in \mathbb{R} : 0 < r < 1$$ Each number has an infinitely small probability of being generated. Does ...
0
votes
0answers
35 views

How to calculate the expected loss of a credit card transaction?

I have used an algorithm to calculate the probability of a credit card transaction to be fraudulent. The algorithm outputs a classification (fraud/no fraud) and the probability of each, such that P(...
0
votes
1answer
55 views

$P(Y\leq X)$ where $Y = X^2$?

Given that $X$ is continuous random variables and we know the probability density function and probability distribution of $X$. We have a new random variable $Y = X^2$. We can easily found it's ...
0
votes
0answers
94 views

Binomial-like distribution with shifting probability

Suppose a student answers a question from a teacher, his probability of getting the question right is p, for each successive answer he gets right, p is reduced by r because the teacher decides to make ...
1
vote
1answer
27 views

Conditional and non-conditional probability in Bayesian network

Question came up on an exam and I'm not sure if what I did was correct. I had to calculate $\mathbb{P}(S)$. Formula for calculating probability of that event would be: $$\sum_{W,U,\lnot W,\lnot U}\...
0
votes
0answers
34 views

Downside covariance?

Is there a notion of downside covariance? And if there is, then I presume that downside covariance plus upside covariance is NOT covariance? If I define the downside covariance (is there an ...
0
votes
1answer
39 views

What is the average and variation of $20$ dices?

If I roll a dice the average is $E(X) = (1+2+3+4+5+6)/6 = 7/2$ and $$E(X^2) = (1+4+9+16+25+36)/6 = 91/6$$ $$VAR(x) = E(X^2) - (E(X))^2 = 91/6 - 49/4 = 35/12$$ Now the question is: How I can find ...
1
vote
2answers
47 views

Probability of N unrelated events, each with different probabilities, what is the chance X number of outcomes occur

Given the probability of N unrelated events, each with different probabilities, what is the chance X number of outcomes occur? Said specifically there are 8 unrelated contracts, what is the chance a ...
1
vote
1answer
47 views

Probability with coins

I'm self learning and I stumbled upon the following task, but I struggle to find the solution: Two players flip coins. The first player flips 3 coins, the second player flips 2 coins. The player that ...
1
vote
1answer
37 views

Probability of picking balls with same color with replacement and without replacement

This is one of our probability exercise: We have an urn with m green balls and n yellow balls. Two balls are drawn at random. What is the probability that the two balls have the same color? (...
0
votes
0answers
39 views

Distribution of sum of squares of dependent but covariance=0 random normal variables

I'm doing some thesis work on lattices, but probability theory is not my strong suit and I'm not sure how to solve this problem: I have vectors $\mathbf{a}_i, \mathbf{x} \in \mathbb{R}^n$, and ...
0
votes
1answer
33 views

Probability Question - Paper Notes in a bag

I could do with some help with this question. In a bag there are 18 paper notes. On five of them there is the digit 2, on seven the digit 3, and on six the digit 5. A man takes 3 notes by random. If ...
0
votes
0answers
17 views

Bernstein-type inequality for simple random walk

Let $(X_n)$ be a sequence of random walks: $P(X_i=1) = P(X_i=-1)=1/2$. Denote $S_n = X_1+...+X_n$. Show that, for $0 < \epsilon \leq 1/4$ $$P\bigg\{\bigg | \frac{S_n}{n} \bigg| \geq \epsilon \bigg\}...
2
votes
2answers
80 views

Find $\mathbb P (X_1 + \cdots + X_n = 6n-3)$

A fair die is tossed n times (for large n). Assume tosses are independent. What is the probability that the sum of the face showing is $6n-3$? Is there a way to do this without random variables ...
2
votes
2answers
71 views

Probability: breaking keyboard

I'm trying to self-learn theory of probability, I came across the following basic problem that I think I solved but I'm not sure as I'm very new to this. Problem: A keyboard manufacturer states that ...
0
votes
1answer
36 views

How can probability and expected value be same in the limit?

I have seen similar arguments at other places but couldn't convince myself so far about it. I am reading some literature related to graph theory but let me post the analogous problem which avoids the ...
0
votes
1answer
23 views

What is the value of $P$?

I have a probability problem. $\\\\\\\\$the problem A class of $100$ students, $42$ studied mathematics, $68$studied psychology, $54$ studied history, $22$ studied both mathematics and history, $25$ ...
0
votes
1answer
34 views

Randomly throwing darts at a dartboard [duplicate]

I have $x$ darts that I randomly throw on a dartboard with $y$ slots (the v shaped slices like you get when you cut a cake). Now I select a particular slice on the dartboard . What's the probability ...
0
votes
0answers
12 views

Is this probability density function (PDF) estimation, $\hat f_{X:\mathcal{X},\theta^*}$, optimum?

Questions: Q1: Is PDF $\hat f_{X:\mathcal{X},\theta^*}$ an optimal estimation of PDF $f_X$? Q2: What are the conditions that, if met, $\hat f_{X:\mathcal{X},\theta^*}$ is optimum? E.g. would it ...
1
vote
1answer
43 views

Shuffles a deck of r ranks and k suits; looks for m cards of the same rank dealt in a row

This is a question that has arisen in my work. Game the First: the dice game. Suppose I have a fair die with r faces. Given some k, I roll the die r*k times; call that sequence of rolls a game. ...
0
votes
2answers
42 views

Order Statistics Intuition

I'm having trouble understanding the intuition behind the answer to a practice question. An internet company has three redundant servers for its web site. Thus, the site functions properly as long as ...
2
votes
3answers
26 views

Characteristic Function of a Conditioned Random Varaible

Let $(X_j)$ be iid random variables and $N \sim \mathrm{Poisson}(\lambda)$, independent of $X_j\, \forall j$. Define $S_n:=\sum_j^n\,X_j$ and consider $S_N$. Find the characteristic function of $S_N$....
1
vote
1answer
30 views

Knowing that the probability to drill success (produced) exploratory hydrocarbon well is $0.2$ by certain company

Knowing that the probability to drill success (produced) exploratory hydrocarbon well is $0.2$ by certain company, this company has drilled four success exploratory wells in different areas on the ...
0
votes
1answer
22 views

MLE in introductory probabilistic information theory

Consider sending a bit that is either $\{0,1\}$ through a noisy symmetric channel, such that for a given input $x$ and a given (potentially noisy) output $y$, $\forall i,j \in \{0,1\}. P(y = i | x = j)...
1
vote
1answer
23 views

Probability of Getting a Yahtzee of Fives Given Two Fives

(The following problem is from MAML, Meet 3, Round 1, December 2012, Problem 3.) In the game of Yahtzee one has a chance to get Yahtzee (5 of the same kind, such as 5 sixes) in the throw of 5 ...
0
votes
0answers
21 views

Probability of getting a “full house” by rolling dice [duplicate]

In poker, full house means getting three cards with the same rank, and another two cards with the same rank (not the same as other three cards). I can understand how to use combination to solve this ...
-1
votes
1answer
67 views

Let $X \sim\operatorname{unif} (1,2)$. Find the distribution of $ Y=X+2/X $

If $ X $ follows the uniform distribution in $ (1,2) $ what is the distribution of $ Y= X + \frac{2}{X} $ ? I thought that $ P( X + 2/X <=y )$ => $ P(X^2-2xy +2 <=0)$ , where y is at $(2\sqrt{...