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 ...
0
votes
1answer
22 views
Decomposing flows on a graph as a sum of cycle flows and source flows
I am reading a paper where they say the following is "easy" but I can't seem to see why.
Let $G$ be a finite undirected graph on an edge set $V$ and
let $E$ be its set of oriented edges (i.e. each ...
1
vote
1answer
27 views
Compute dependent probability
I have
$X = \pmatrix{-2 & -1 & 0 & 1 & 2 \\
.05 & .2 & .3 & .4 & .05} $, $$F(X) =
\begin{cases}
0 & X \le -2 \\
.05 & -2 \le X < -1 \\
.25 & -1 \le X ...
0
votes
0answers
11 views
Algorithm for Geometric Distribution
Not a long time ago, I asked what distribution the following algorithm would lead to for the x's:
for i=1:n
k = 0;
...
0
votes
1answer
21 views
How to do I calculate the conditional probability distribution?
The Chicago Cubs are playing a best-of-five-game series (the first team to win 3 games win the series and no other games are played) against the St. Louis Cardinals. Let X denotes the total number ...
0
votes
2answers
50 views
For $X_i$ iid $ Var( \sum_{i=1}^n X_i )= \sum_{i=1}^n Var (X_i)$?
My contention is that it's true. I thought of two ways of proving it, unsure which one is better (and/or correct):
Suppose $X_i \sim N(\mu, \sigma^2)$, then it's moment generating function (MGF) is
...
2
votes
3answers
76 views
Probability of winning the game “1-2-3”
Ok, game is as follow, with spanish cards (you can do it with poker cards using the As as a 1)
You shuffle, put the deck face bottom, and start turning the cards one by one, saying a number each time ...
0
votes
1answer
20 views
Probability question regarding $3$ pairs of boxes with different objects
Suppose there are two boxes each containing an eraser, another two boxes each containing a pencil, and another two boxes each containing a pen. The boxes are not labelled, and we need at least one ...
0
votes
1answer
46 views
what is the probability that $y\le \sin x$?
We need to find in a rectangle $[0,{\pi\over 2}]\times [0,1]$ a point $(x,y)$ is chosen at random what is the probability that $y\le \sin x$?
Will it be $${\int_{0}^{\pi/2} \sin x \ dx\over \pi/2}$$
1
vote
2answers
25 views
Is There a Better Strategy for this Combination Scenario?
An elevator containing five people can stop at any of seven floors.
What's the probability that no two people get off at the same floor?
Assume that the occupants act independently
that all floors ...
1
vote
2answers
48 views
Find the probability $P(0.5 < X < 5)$
We select two balls without replacing from a box where there are 7 red balls and 3 green balls.
Be $X$ the random variable denoting the number of selected green balls.
Please compute $P(0.5 < X ...
1
vote
0answers
36 views
Conditional probability on multiple conditions
I want to calculate probabilities like
$$P(B_1=1,B_2=0,B_3=0\mid C_1=1, C_2=1, C_3=0)$$
I know the separate pairwise conditional probabilities like
$P(B_1=1\mid C_1=1)$ for all $B_1,\ldots,B_n$ and ...
1
vote
2answers
44 views
why is this Markov Chain aperiodic
I have this Matrix:
$$P=\begin{pmatrix} 0 & 1 \\ 0.3 & 0.7 \end{pmatrix}$$
this markov chain is said to be aperiodic, I dont understand how it comes to it. Period $\delta$ is the gcd of ...
7
votes
2answers
369 views
the following inequality is true, but I can't prove it
The inequality $$\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)$$ holds for all integer $d\geq 1$. I use computer to verify it for $d\leq 50$, and find ...
1
vote
0answers
25 views
Hypergeometric Distribution Probability (mean, variance, Std Deviation)
The distribution of the number of children per household for households receiving aid to dependent children (ADC) in a large eastern city is as follows: 5% of ADC households have one child, 35% of ADC ...
5
votes
3answers
53 views
Help with conditional probability?
I've got to show that:
$$\mathbb{P} (A | A \cap B) = \frac{ \mathbb{P}(A)}{ \mathbb{P} (A \cap B)}$$
I'm not sure how to get to this.
Surely the probability of A occurring given A and B occurs is ...
0
votes
1answer
44 views
Proof that De Moivre-Laplace theorem is a particular case of the Central limit theorem
Hello everyone I have this problem Can somebody help me please
Proof that De Moivre-Laplace theorem is a particular case of the Central limit theorem
(remember that a binomial is the sum of ...
0
votes
1answer
24 views
I need help with a specific probability question
I have a probability question. The digits 1, 1, 1, 1, 2, 2, 8, and 8 are each written on a piece of paper and placed in a bag. The pieces of paper are randomly drawn one at a time. The digits are ...
0
votes
2answers
29 views
law of large numbers, convergence of random variables, probability distribution, and a lottery
The Gambler's Fallacy tells us we cannot predict the next coin flip result based upon history. Several heads in a row do not mean the next flip is any more likely to be tails. However, the law of ...
1
vote
1answer
57 views
Can you use combinatorics rather than a tree for a best of five match?
The Chicago Cubs are playing a best-of-five-game series (the first team to win 3 games win the series and no other games are played) against the St. Louis Cardinals. Let X denotes the total number ...
0
votes
0answers
34 views
Integrable local martingale is a supermartingale
Let $M_t \in L^1(\mathbb P)$ be a local martingale. Hence exists an increasing sequence of stopping times $\tau_n$, for each of which the process $M_t$ is a martingale.
\begin{align}
\mathbb E [M_t ...
0
votes
1answer
63 views
A very strange deck - probability and expected number of draws
Say we have a virtual deck of 70 cards of four suits and each player has access to his/her own unique independent deck (one players' actions do not affect another player's):
$$
\begin{array}{r|rr}
...
1
vote
1answer
26 views
Distribution of the number of heads in a simple coin tossing process with 2 coins.
You have 2 unfair coins, A and B. Consider the following process:
1. toss A
2. If it's head go to 1
3. toss B
3. if it's head go to 1
What is the distribution of the number of heads if then ...
0
votes
1answer
54 views
Understanding Chebyshev's Theorem
I'm reading about Chebyshev's Theorem and I am seeing conflicting information. My book states:
Chebyshev's Theorem is a general result that applies to most discrete random variables (and most ...
0
votes
1answer
30 views
How do you invert a characteristic function, when integral does not converge?
I need to find the probability density of some distribution with characteristic function given by:
$$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$
I know the formula for inverting a ...
0
votes
0answers
14 views
Smoothing technique for parameter estimation
I have a real-world web-graph and am trying to check the formula
$P = cd^{-\gamma}$, where $d$ is the degree.
I have a problem that there are too many verticies with unique d, so one cannot calculate ...
0
votes
1answer
25 views
Unbiasedness of product/quotient of two unbiased estimators
An answer to this question might just be "it depends", however I am wondering:
Given unbiased estimators $\hat{\mu}_X$ and $\hat{\mu}_Y$ for the means $\mu_X$ of $X$ and $\mu_Y$ of $Y$ respectively. ...
1
vote
0answers
37 views
Implicit differentiation of differential equation
Let the unknown cdf $F(x)$ be implicity defined by
$h(F(x);a,b) := F(x)[1-a-b(1+a)] - 2 a b F'(x) x + (1+a)b = 0$, where
$F(1) = 1$.
Moreover, let $0<a<1$, $0<b<1$.
My question is: is ...
0
votes
0answers
27 views
Solving Probability Balance Equations Matrix
I have a really simple question to ask which for some reason I just cannot get around!
Say we are given equations
$a_1 = a_2 p + a_3 q$
$a_2 = a_1 p + a_3 q$
$a_3 = a_1 p + a_2 q$
Taken from the ...
1
vote
3answers
33 views
Probability to select all 3 male mouses from 10 selected at random
In a cage there are 100 mouses from which 3 are male.
Compute the probability of selecting all 3 males from a group of 10 mouses selected at random.
I have this intuition:
$$
P(male)=0.03
$$
and ...
-2
votes
0answers
36 views
Probability distribution function and degenerate distribution
Random variable X is equal in distribution to 0 with probability p and to Poisson(m) with probability 1-p. P(X=k)=?
0
votes
0answers
36 views
How to calculate $\int^{\infty}_{-\infty}\Phi(\frac{w-a}{b})\phi(w)dw$, where $\phi$ and $\Phi$ are the standard normal density and distribution
Suppose $\phi(\cdot)$ and $\Phi(\cdot)$ are density function and distribution function of the standard normal distribution.
How can one calculate the integral:
...
1
vote
0answers
29 views
Why can't I use the variance of the sample average in the Central Limit Theorem for the weak-stationary process?
Under mild conditions $\dfrac{\bar{X}-\mu}{\sqrt{\sigma^2/n}}$ approaches the standard normal (where $\sigma^2$ is the process variance, not the marginal variance $\sigma^2_x$).
Why is the ...
0
votes
0answers
39 views
Occupancy Problem in Game Theory
I try to solve a problem that is defined as the mix of combinatorics and game theory and very similar to the occupancy problem.
Problem : There are $n$ cells and some number of balls. At each round, ...
1
vote
1answer
27 views
Probability Theory Question
A manufacturer makes 3 models of a TV, models A, B and C. A store sells 40% model A sets, 40% model B sets and 20% model C sets. Of model A sets, 3% have stereo sound; of model B sets, 7% have stereo ...
1
vote
1answer
29 views
Probability Theory (Pinochle deck of cards)
-=Attempts added=-
A Pinochle deck is a special deck of cards with 48 cards in total. it consists of two copies of each of the 9, 10, J, Q, K and Ace of all four suits (so there are 2 nine of clubs, ...
0
votes
1answer
31 views
How find this In any group of $2n-10$ persons,there are always at least $10$ persons who have the same birthdays.
What is the smallest posiible integer value of $n$ such that the following statement is always true
In any group of $2n-10$ persons,there are always at least $10$ persons who have the same ...
0
votes
1answer
30 views
How Would You Interpret This Question?
A man has 5 coins, two of which are double-headed,
one is double-tailed, and two are normal.
He picks a coin at random and tosses it.
What's the probability that the lower face is a tail?
P(H) = ...
1
vote
1answer
12 views
probability hypergeometric distribution
An HR manager estimates that 35% of married employees in a large office complex have spouses whose employers provide dental insurance and 65% have spouses whose employers provide extended medical and ...
1
vote
2answers
43 views
How to derive the expected value of $X^\alpha\log X$
Let $X$ follow a Weibull distribution, with density $$f(x)=\frac{\alpha}{\theta}x^{\alpha-1}e^{-\frac{x^{\alpha}}{\theta}}\quad x>0 .$$
How can I find the following expectation?
...
3
votes
0answers
42 views
Convergence of a Subordinator.
Let $\left( X_{t}\right) _{t\geq0}$ be a subordinator with the Laplace
expoent given by
$$
\Phi\left( \lambda\right) =d\lambda+\int_{0}^{\infty}\left( 1-e^{-\lambda
x}\right) \nu\left( ...
2
votes
4answers
32 views
Mutually Exclusive Events (or not)
Suppose that P(A) = 0.42, P(B) = 0.38 and P(A U B) = 0.70. Are A and B mutually exclusive? Explain your answer.
Now from what I gather, mutually exclusive events are those that are not dependent upon ...
1
vote
1answer
17 views
probability of who will be selected
"In an office there are 3 secretaries, 4 accountants, and 2 receptionists. If a committee of 3 is to be formed, find the probability that one of each will be selected?
Attempted Solution:
First ...
2
votes
2answers
58 views
X,Y are independent exponentially distributed then what is the distribution of X/(X+Y)
Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together.
The exercise goes as ...
1
vote
1answer
30 views
Probability of being matched against a pair of people
$\textbf{Question:}$ Suppose you are playing a game in which two teams of five people, call them Team A and Team B, compete. Each of the ten people is randomly assigned a unique role (no two people ...
1
vote
3answers
42 views
The probability of getting more heads than tails in a coin toss
A fair coin is to be tossed $8$ times. What is the probability that more of the tosses will result in heads than will result in tails?
$\textbf{Guess:}$ I'm guessing that by symmetry, we can ...
3
votes
2answers
27 views
Probability Calculation using combinations
In a population of $250$ items, $20$ are defective. Suppose $4$ items are sampled at random, without replacement.
a. What is the probability that the sample will consist of $4$ defective items?
...
-1
votes
0answers
28 views
Uncertainty of adopting options in medical treatment. [closed]
A person needs to be near a hospital to receive treatment.
She answered a survey on the premise be telling the TRUTH.
Option A - She says she's close to the hospital. (treatment is accepted)
Option ...
0
votes
1answer
30 views
finding unconditional distribution by integrating conditional distribution
Given $$ f_Y (y)= \begin{cases}\frac{1}{120} e^{-\frac{1}{120}y} &, y\ge 0 \\ 0, &, y< 0 \end{cases}$$
and
$$f_{X|Y} (x|y) = \begin{cases}\frac{1}{y} &, x\in [0, y] \\0 &, ...
3
votes
2answers
66 views
Mean of an increasing function over exponential distribution
I came across the following problem in my research
I have two random variables $X, Y$ which are exponentially distributed and $Y$ has a higher mean than $X$.
Then I have a function, say $f(z)$, ...
1
vote
0answers
30 views
Expectation of $X$log$(X)$ when $X$ is Poisson random variable
I am trying to compute $\mathbb{E}[X\mathrm{log}(X)]$ where $X$ is a Poisson random variable with mean $\lambda$, so $Pr(X=x) = \frac{e^{-\lambda}\lambda^x}{x!}$. My usual approach to computing ...
