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
3answers
64 views

Probability of a year which is not a leap year

If a 4 digit year is choosen randomly, what is the probability that it is not a leap year ? This problem has come in my exam and i have written like this I know that the number of four digit year ...
1
vote
2answers
37 views

what is the distribution of a uniform r.v. divided by an exponential r.v.?

Say $U=\frac{X}{Y}$. X and Y are independent with each other. X is a Uniform distribution r.v. $X\sim \mathcal{U}(0,1)$. Y is an exponential distribution r.v., $Y\sim\mathcal{Exp}(\lambda)$, whose pdf ...
0
votes
1answer
61 views

Probability and coin tosses

Taking a Probability & Statistics class this term and trying to get my head wrapped around how I calculate coin tosses with specific out comes in mind. We're using the nCr and nPr functions on our ...
1
vote
3answers
39 views

Determine the law of $F^{-1}(U)$, $U$ uniformly distributed on $[0,1]$

i'm trying to understand the following problem Let $X$ be a real random variable, its distribution function is $F(t):\Bbb{P}(X\le t), \forall t\in \Bbb{R}$. Define the right-continuous inverse by ...
2
votes
2answers
56 views

conditional probability about sum and product rule

I am reading Bishop's Pattern Recognition and Machine Learning. In page 73, chapter 2.1. I can't understand the formula 2.19 : $$p(x=1|\mathcal{D})=\int_0^1 p(x=1|\mu)p(\mu|\mathcal{D})\text{d}\mu ...
0
votes
0answers
26 views

What is the optimal prize for a prize ticket in a raffle [on hold]

What, if any is the optimal price for a prize ticket given the value of a prize? For example if you were to raffle a TV and wanted to cover the cost of the prize? Let say the people were aware of how ...
0
votes
2answers
29 views

conditional probability maybe?

If in application A, 70% of the users are men and 30% women. In application B, 60% men and 40% women. Given you have both applications, what is the probability that you are a man?
-2
votes
1answer
32 views

Baye's theorem may be required. [on hold]

A message is sent which consists of $n$ binary symbols $0$ and $1$. Each symbol is distorted with a small probability $p$ (is changed to the opposite). To be on the safe side the message is repeated ...
0
votes
0answers
35 views

Kelly criterion for 3 outcomes

I have been exploring the Kelly criterion for optimizing the bet size for a two outcome bet situation. I'm having trouble applying this to a three outcome bet. I may refer to this excellent thread: ...
-2
votes
1answer
32 views

Expected Value Question Intermediate [on hold]

Mila has four ropes. She chooses two of the eight loose ends at random (possibly from the same rope) and ties them together, leaving six loose ends. She again chooses two of these six ends at random ...
-2
votes
1answer
24 views

Binomial Probability help [on hold]

The problem is: $35\%$ percent of the employees in a company receive an incentive in the month of April. What is the probability that $4$ employees of the company chosen at random do not receive the ...
1
vote
1answer
40 views

Sum of truncated normal random variable and normal random variable

I'm wondering if there is a closed-form pdf of sum of "correlated" normal random variable and truncated normal random variable. I found a paper providing the pdf for "uncorrelated" case, but could ...
0
votes
2answers
44 views

Check if a given function is a probability density function [closed]

Given $f(x)=\tfrac1{π(1+x^2)}, ~x\in(-\infty, \infty)$, is it true that $f$ is the probability density function of some continuous random variable?
2
votes
2answers
73 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
2
votes
1answer
38 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
1
vote
1answer
37 views

Kids and cookies, probability

10 children $D_1,D_2,...,D_{10}$ were given 20 cookies. What's the probability that $D_{10}$ has at least one cookie if we know that $D_1$ and $D_2$ both have exactly 2 cookies. I think that by ...
0
votes
1answer
49 views

Fast way to inverse B'CB+D

$\mathbf {A = B'CB}$, where $\mathbf A$ is of dimension $n \times n$, $\mathbf C$ is m by m, positive definite and symmetric, $\mathbf B$ is of dimension $m \times n$, and $n >> m$. Inversion ...
3
votes
1answer
43 views

Statistical test for “too perfect” random number generator?

I am attempting to characterize some random number generator programs in a very simple way. Specifically, I'm rolling a simulated 6-sided die $3 \times 10^8$ times and keeping a count of how many ...
0
votes
1answer
22 views

Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
0
votes
1answer
25 views

conditional probability of throwing a dice

I would like to compute the conditional probability of throwing a dice. The event $A$ is getting 2 and the event $B$ is the number to be even, so the question is what is the probability of getting 2 ...
5
votes
4answers
131 views

What is the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$ and some of the properties of $E[X \mid Y]$?

I was trying to understand both intuitively and rigorously what the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$. Let me tell you first the things that do make sense to me. $E[X\mid Y=y]$ makes ...
1
vote
1answer
25 views

Probability of multiple variables, geometric distribution?

You are on a basketball team, and at the end of every practice, you shoot half-court shots until you make one. Once you make a shot, you go home. Each half-court shot, independent of all other shots, ...
0
votes
0answers
36 views

The mathematical odds of winning a hand in poker with two boards

In Hold-em, after the flop, one hand has two pairs and the other hand has a flush draw. The odds of two pairs winning against a flush draw after the flop with 2 cards to come is roughly 3:2. In ...
1
vote
1answer
36 views

Intersection of countable many sets of measure $1$

Consider a probability space $(X,\mathscr M,\mu)$ and a collection of measurable sets $\{A_n\}_{n\in\mathbb N}$ such that $\mu (A_n)=1$ for every $n$. Then I don't unterstand the following result: ...
6
votes
1answer
68 views

Probability that two circles in space are linked

Let $C_0$ be a circle centered on the origin, and $C_1$ a circle centered on $(1,0,0)$, center distance of $1$. Q1. If both $C_0$ and $C_1$ are randomly oriented and have the same radius $r ...
-1
votes
3answers
68 views

The chance to double 1000 points into 2000 points [closed]

You own 1000 points. Your goal is to reach 2000 points, the only way you gain points is by gambling. You will always gamble 40 points, your chance of winning a 40 points gamble is 60%, how high is ...
0
votes
1answer
24 views

How do you get the probability distribution of the sum of random variables by using the inverse of the transform?

I read the following statement: If X and Y are independent random variables, the distribution of their sum W = X + Y can be obtained by computing and then inverting the transform $M_W (s) = ...
1
vote
2answers
56 views

Coin flip gamble

You have an amount of money to bet on a fair coin flipping and landing on heads. How much should you bet as a function of your balance to maximize your probability of profiting if you play $x$ times?
1
vote
2answers
53 views

let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous.

i'm trying to understand a proof of the following statement: let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous. I'll write down the proof in such a ...
4
votes
2answers
163 views

Drunken sailor's Random Walking

A drunken walker is on $x=0$, and if $x<0$, he falls and he dies.(Once he gets position $x<0$, he dies permanently.) There is $0<p<1$ chance to move right ($x \rightarrow x+1$), and $1-p$ ...
1
vote
1answer
54 views

Probability of heads given we observe HTH?

I am observing a sequence of heads and tails and trying to deduce the bias of my coin. Let's say I observe HTH. Can I estimate the bias of my coin $p$ ? ...
1
vote
1answer
8 views

Multivariate hypergeometric distribution

The question asks: 20 individuals consisting of 10 married couples are to be seated at 5 different tables, with 4 people at each table. If 2 men and 2 women are randomly chosen to be seated at each ...
0
votes
2answers
32 views

Calculating the probability of getting a specific set of dice from a single roll

If I want to create a function $p_{roll}(x_1, x_2, x_3, x_4, x_5)$ that calculates the probability of getting the given dice (order is not important) in a single roll with 5 dice then how would that ...
0
votes
0answers
27 views

Proving convergence of events [closed]

Let $A_1,A_2,\dots$ be events. Prove that $A_n\rightarrow A$ if and only if $I_{A_n}\rightarrow I_A$ as functions on $\Omega$, that is, $I_{A_n}(w)\rightarrow I_A(w)$ for every $w\in\Omega$. $I_B$ ...
-1
votes
1answer
31 views

Prove the inclusion-exclusion principle by probability [closed]

Let $A_1,\ldots,A_n$ be events, and for $J\subseteq \{1,\ldots,n\}$, let $B_J = \cap_{j\in J}A_j$ . For $k\ge 1$, let $S_k=\sum_{|J|=k}P(B_J)$, where the sum is over all subsets $J$ of $\{1,...,n\}$ ...
-1
votes
1answer
37 views

Prove that the set function is a probability [closed]

Suppose $P_1$ and $P_2$ are probability measure on measurable space $(\Omega,{\mathcal F})$ and also $\alpha$ is a real number such that $0 \le α \le 1$. Prove that the set function $P(A) = \alpha ...
-1
votes
0answers
16 views

what are the joint distribution functions and copula? [closed]

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ A &=&\frac{U+V}{2}, \\ G &=&\sqrt{UV}, \\ H ...
1
vote
0answers
63 views

Fast Convergence of marginal distribtution

Let $(q_n)$ be sequence of probability density functions of the couple $(x,y)\in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. $p_n(x):=\int q_n(x,y)dy$. Another sequence of functions ...
3
votes
1answer
28 views

countably additive function P

This problem comes from exercise 1.3.5(b) of 'A First Look at Rigorous Probability Theory'. It asks to give an example of a countably additive function $P$, defined on all subsets of $[0,1]$, which ...
-3
votes
0answers
44 views

Party-dance-boys-girls [closed]

At a party,there are $n$ boys and $n$ girls.Each boy danced with exactly $k$ girls and each two boys danced with exactly $d<k$ common girls.Show that each girl danced with exactly $k$ boys and each ...
1
vote
1answer
66 views

Probability Density function help… $(X,Y) \sim f(x,y)=\frac{1}{x}$ where $I = (0 < y < x < 1 )$ …

a) show that $f(x,y)$ is a probability density function. b) $f_Y (y\mid X=x_0) =$ ? (for what $y$ is this the correct "formula"?) c) $f_Y (y) = $ ? My ideas: a) Clearly in our $I$ this is positive ...
1
vote
2answers
67 views

Probability Help! X and Y are geometric RV's with parameter p. What are …

a) $P\{X + Y = n\} (n = 1,2,...)$? b) $P\{X = k | X + Y = n\} (1 ≤ k < n)$? My work thus far... a) $\begin{align}P(X + Y = 1) & = P((X = 0 \cap Y = 1) \cup (X = 1 \cap Y = 0)) \\ & = ...
0
votes
2answers
148 views

We toss three coins (each with pr(heads)= p). Let X be the number of heads that occur on the first two tosses and Y be the number of heads..

that occur on tosses 2 and 3. range of X = range of Y = {0,1,2} Does this work seem at all correct? I am stumped with this problem... I'm not sure how to approach it. Any help is greatly ...
1
vote
1answer
82 views

Probability Help! (X,Y) ~ f(x,y) = 8xy $I_D(x,y)$

a) $f_X (x) =$ ? b) $P( X + Y < \frac{1}{2}) =$ ? c) $f_Y(y \,| \, X = \frac{3}{4}) =$ ? d) $P( Y < \frac{1}{2} \, | \, X = \frac{3}{4}) = $ ? Any help is greatly appreciated! Thanks!! Here ...
0
votes
1answer
17 views

Is there a way to use the Generalized Mean to find the “best” possible mean to use for a specific data set?

I've recently learned about the Generalized Mean as an abstraction of the many different means, includeing the Geometric, Arithmetic, and Harmonic means, as well as others. It is my understanding ...
1
vote
1answer
117 views

Conditions for convergence of moments

Let ${X_n}$ be a sequence of r.v. such that $X_n\xrightarrow [d]{}X$, with $E(X)$ finite, and with $E(|X_n|^{1+\delta})\leq K<\infty$ for all $n$. We know that: a) For $\delta>0$, we have ...
0
votes
1answer
36 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
0
votes
0answers
61 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
0
votes
3answers
34 views

Problem on Probability

What is the proportion of numbers between $100$ and $999$ that are not divisible by $7$? please tell us the shortest method to find that kind of problems.
0
votes
0answers
56 views

Share oranges evenly

There is a boy in a street sharing his oranges with people coming across to him on a street. He has 100 oranges in his basket. The number of people walking toward the boy are unknown and varies in ...