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 ...

learn more… | top users | synonyms (2)

0
votes
1answer
12 views

Drawing marbles out of a bag with or without replacement

Suppose the marble population has $n$ marbles. And based on prior knowledge I know that 30% of them are red, 50% of them are green, and 20% of them are blue. If I want to sample 2 marbles from my ...
-2
votes
1answer
14 views

Trinomial Distribution - Cumulative Probability of (X-Y)

In an election there are n voters. They can each vote for Candidate A (with probability p); Candidate B (with probability q) or neither (with probability (1-p-q) ). What is the Probability that ...
1
vote
0answers
23 views

Probability calculation involving order statistics

There are $n+1$ iid random variables with the same distribution as $D \sim \text{Exp}(\mu)$, denoted by $D, D_1, D_2, \ldots, D_n$. Define event $E_1$ as "$D$ is less than the $(\lfloor n/2 \rfloor + ...
0
votes
2answers
14 views

Find constants using the mean, variance and covariance of two random variables

Given two random variables $X$ and $Y$ such that $\mathbb{E}(Y\mid X)=a+bX$, find $a$ and $b$ in terms of the mean, the variance and the covariance of $X$ and $Y$. Hint: What is the relationship ...
2
votes
1answer
19 views

Find the probability generating function $G(s)$ of this branching process.

Suppose that $X_n$ is size of the $n$th generation of a branching process started from a single individual, where each individual has a random number of children with probability mass function: ...
-3
votes
1answer
23 views

The Crawfords have $3$ children — questions about their probability distribution [on hold]

Suppose the Crawfords have three children. Assume the probability of a boy or a girl is $\frac12$ for each birth. How many possible outcomes are there? What is the probability that only two of the ...
-3
votes
1answer
26 views

Time taken to give answer if probability is given.

This is a question that I am struggling with: Since the password is periodically changed, you would like to know the answer as soon as possible. So you decide to interrogate the minions in an order ...
1
vote
2answers
17 views

Probability of Independent Events individual vs in series

I understand that independent events (such as a fair coin flip) should not be viewed in succession. For example, if you flip heads 10 times in a row, the odds of flipping the next coin heads is still ...
1
vote
1answer
26 views

Show that the empty set is independent of $A$ for any $A$

I am somewhat stumped as to how to approach this. The only thing I can remotely think of is $$P(A\cap \emptyset) = P(\emptyset)$$ but nothing else comes to mind. Suggestions?
2
votes
0answers
11 views

Convergence of Sum of Random Variables “Independent in Limit”

Consider a sequence of random variables $X_n\sim U[-n,n]$, a random variable $Y\sim N(0,1)$, and a random variable $Z\sim U[0,1]$, all independently distributed. In addition, consider a bounded, ...
1
vote
1answer
18 views

Suppose that $N$ is an iid geometric RV and $X_i$ is an iid Bernoulli RV. Find the p.g.f. of $R=X_1+ \dots + X_n$.

Each year a tree of a particular type flowers once and produces a random number $N$ of flowers, where $\mathbb{P}(N=n)=(1-p)p^n$, $n=0,1,2,\dots $ and $0<p<1$. Each flower has probability $1/2$ ...
4
votes
0answers
41 views

How many possible shuffles can be won perfectly?

It is known that the possible shuffles of a deck of cards is $52!$, or ~$80658175170943878571660636856403766975289505440883277824000000000000$ different combinations. I have become aware of a game ...
0
votes
0answers
13 views

Expectation Value and Generalized Holder Inequality

In the context of probability, I need help in interpreting a generalized Holder inequality (wiki): $\| \prod_{k=1}^n f_k \|_r \leq \prod_{k=1}^n \| f_k \|_{p_k}$, where $\sum_{k=1}^n \dfrac{1}{p_k} ...
0
votes
1answer
24 views

Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :

$$P(X_1 < X_2 < X_3)= \frac{(1-p_1)(1-p_2)p_2p_3^2}{(1-p_2p_3)(1-p_1p_2p_3)}$$ To be frank I do not know where to start with this question, I would like an idea to get me going, or better yet ...
2
votes
1answer
14 views

Convergence in probability of product random variables

If $Y_n$s converge to constant $c$ in probability & $(X_n)$ is a sequense of random variables, is it true that $X_nY_n- cX_n$ converge to $0$ in probability? How can I prove this? Thanks in ...
2
votes
2answers
25 views

Conditional probability calculation (multivariate distribution)

X and Y are two i.i.d. random variables having the uniform distribution in $[0,1]$ The question is to calculate $Pr(Y\geq \frac{1}{2} | Y\geq 1-2X)$ My calculations: $$ \begin{align} Pr(Y\geq ...
0
votes
1answer
19 views

How to derive the covariance formula

In my book, I am given this proof: $$ Cov(X,Y) = \mathbb{E}[X - \mathbb{E}X][Y - \mathbb{E}Y]$$ $$ Cov(X,Y) = \mathbb{E}[XY] - 2\mathbb{E}[X] \mathbb{E}[Y] + \mathbb{E}[X] \mathbb{E}[Y]$$ I do not ...
1
vote
1answer
22 views

multinomial hypothesis testing

Suppose we have data $(X_1, X_2, X_3)$ (I'll refer the categories as 1, 2, 3) that has a multinomial distribution with parameters $n$ and $(p_1, p_2, p_3)$ and we want to test the hypothesis that ...
2
votes
2answers
37 views

Probability and Combinatorics without replacement

If I have a sample space of $A$ and I randomly select $a$ elements, mark them, put them back into the sample space, then randomly select $b$ elements and I want to know what the probability is that ...
0
votes
1answer
19 views

Expected time of drawing all types of coins from a large pile

I've been working on the following question but am uncertain of how to solve it Consider an infinitely large pile of coins. Each coin has a number {1, 2, . . . , n} written on it, and these numbers ...
2
votes
0answers
28 views

An exercise on conditional expectation and some related questions.

I tried to solve an exercise involving conditional expectations, and in doing so some question's popped up in my mind. First the exercise: $|Z| \le c \textrm{ P.-a.s.} \Rightarrow |E\{ Z | ...
0
votes
2answers
28 views

A stick length 1 is broken into 2 pieces. Let $Z_1$ be the length of the shorter part. Find $EZ_1$

This is used: If $p(x)$ is continuous, then $P\{x \leq X \leq x+ \Delta x \}= p(x)\Delta x+ o(\Delta x), \Delta x\to 0.$ Let $H_1$ be the occurrence that the point at which the stick is broken is in ...
-1
votes
1answer
27 views

Find the probability that this laptop is produced by the firm B. [on hold]

The store received three laptops of different firms (firm A, firm B and firm C) in a ratio of 6:1:3. Laptops coming from firm A do not require repair during the warranty period in $98$ percent of ...
0
votes
1answer
26 views

Finding $E[W]$ and $E[W^2]$, where $W = \int_{t=0}^T B_s$ $ds$

I'm trying to find a)$E[W]$ and b) $E[W^2]$, where $W_t = \int_{t=0}^T B_s$ $ds$ ($B_s$ denotes a Brownian motion). In addition, I'd like to find $E[Z_sZ_t]$, where $Z_t = \int_0^t B_s^2$$ ds$ ...
1
vote
1answer
19 views

Joint probability distribution (over unit circle)

A couple of two continuous random variables $(X,Y)$ is distributed uniformly over the closed unity circle (so $-1\leq x \leq 1$ , $y$ analog). $U$ is defined as the distance from $O$ to the point ...
2
votes
1answer
29 views

Translation:Bayes Classificator -> precise math?

I want to understand the most simple form of the Bayes classificator (see here) but I want to understand it in a really precise, clean, mathematical way. Math description of the setting: Let us ...
0
votes
1answer
22 views

Take the outcome of a draw in ELO formula

Is there any way to get the probability of a draw outcome using ELO formula as it only gives the Win probability ELO formula is given by $E = \frac{1}{1+10^\frac{d}{a}}$ where d is the difference in ...
0
votes
1answer
31 views

Expectation of the product of Brownian motions

I'm new to Stack Exchange. I'd like to find the expectation of the product of three Brownian motions: $E(B(t_1)B(t_2)B(t_3))$ I know from a previous post here that ...
1
vote
0answers
22 views

Why aren't these two sets of stochastic processes equal?

I'm learning about stochastic integrals now, and I don't understand the following: If $S$ and $L$ are two classes of processes where: $S=\{f(s,\omega) |f $ is progressively measurable and ...
-3
votes
3answers
32 views

Two random variables [on hold]

$X$ and $Y$ are two independent identically distributed random variable with pdfs given by $0.5\big[\exp (- \vert x-1\vert) \big]$, where $x$ and $y$ range from $-\infty$. to $+\infty$. Question ...
0
votes
1answer
39 views

Let $X: U(0,3 \pi)$ - Uniform on $(0,3 \pi)$ Find the distribution of Y and the expectation of Y if Y is:

$$Y=\begin{cases} -\sin X , x \in(0, \pi] \\- \frac{1}{2} , X \in[\pi, \frac{3 \pi}{2}]\\ \cos X, X \in [\frac{5 \pi}{2}, \frac{11 \pi}{4}] \\ \frac{3}{4}, X \in (\frac{11 \pi }{4}, 3 \pi) ...
-2
votes
0answers
34 views

Calculating probability [on hold]

I am not a mathematician so I am not sure if I am asking this properly. I wanted to find if semantic meaning could be derived from digital roots of spiritual words. I calculated roots of 76 words and ...
-6
votes
0answers
41 views

probability that a hedgehog safely crosses a road [on hold]

If a hedgehog crosses a certain road before 7.00 a.m., the probability of being run over is $\frac1{10}$. After 7.00a.m., the corresponding probability is $\frac34$. The probability of the hedgehog ...
0
votes
3answers
28 views

Random independent variables, a question of expected value

The density function of $X$ and $Y$ (two independent variables) are respectively : $$\phi_X(x)=\begin{cases} \frac{1}{2}(1+x) , x \in (-1,1) \\ 0, \text{otherwise} \end{cases}$$ ...
0
votes
1answer
13 views

Find the density function of $X$, from the random vector $(X,Y)$ if the PDF of this vector is:

$$\phi(x,y)= \frac{|x|}{\sqrt{8 \pi}}e^{-|x|- \frac{1}{2}x^2y^2}, x,y \in R $$ Now I'm aware I would have to do $$\phi_X(x)=\int_{- \infty}^{+ \infty}\phi(x,y) dy$$, what is confusing me is this ...
2
votes
0answers
33 views

Calculate mean and correlation of a stochastic process

I am given the Stochastic process $Y_n$, where $n \in Z$ defined by: $ Y_n = X_n - X_{n-1}$ where $X_n$ is a process with independent and identically distributed geometric variables $X_n \sim G(p)$ ...
1
vote
1answer
22 views

Definition of random variable

In some books, they don't define the random variable based on measure theory. Instead, they define as follow (in the book All of Statistics of Larry Wasserman): My question is does this definition ...
0
votes
0answers
20 views

How does probability generating functions relate to thinning of a Poisson process?

For my exam in probability one of the questions is: "7. Generating functions (possibly with a focus on how probability generating functions relate to thinning of a Poisson process)." I can't figure ...
1
vote
1answer
35 views

Drawing at least one of each color marble

Suppose I have 300 marbles. 24 of them are red, 59 of them are green, 66 of them are blue, and 151 of them are yellow. What's the probability of drawing at least one of each color after 30 draws? So ...
1
vote
2answers
23 views

Basic question on the probability function and the probability distribution function

I have a question on the probability function. In my book it says that if A and B are mutually exclusive events $P(A∪B)=P(A) + P(B)$. Then when it starts talking about the probability distribution ...
1
vote
2answers
29 views

Sum of Binomial distribution when the success rate is different.

Is there any easy way to calculate the probability of the sum of two binomial random variable if the success rates of them are different each other? I mean that $X \sim Bin(n,p_0)%$, $Y \sim Bin(m, ...
0
votes
1answer
34 views

'Perfect Distribution' probability question [on hold]

If I throw two $6$-sided unbiased dice (with faces $1$ through $6$) thirty-six times, what is the probability that each sum appears exactly according to the $\{1,2,3,4,5,6,5,4,3,2,1\}$ distribution?
0
votes
1answer
30 views

Book on probability theory with sigma algebra

Please suggest or recommend a book on Probability theory emphasising on sigma algebra and with Kolmogorov’s axiomatic development.
0
votes
0answers
27 views

Selecting Keys From A Basket [on hold]

Nine women and two men sat in chairs. A male occupied seat 9 and 11. Keys were drawn from a basket in the order they sat. Find the probability of the woman in the sixth seat selecting the correct key ...
0
votes
2answers
24 views

Probability of picking 2 red and 2 blue marbles from a bag of 5 red and 5 blue

You pick 4 marbles from a bag of 5 red and 5 blue, with no replacement. What is the probability of getting exactly 2 marbles of each color? I think it's either 3/8 or 10/21 but I'm not sure which.
2
votes
1answer
52 views

Can someone help explain a proof from Feller Vol1 III.5?

One will need a copy of Feller's text (3rd edition) to answer this question. The proof I'm having difficulty with is Theorem 1, pages 84-85. When he discusses the r=1 case, he says ... "To the ...
0
votes
1answer
16 views

Sum of iid random variables with an odd distribution

I have $G_1,G_2$, iid with probability distribution function $f(y) = Ce^{-y}y^{-1/2}$ where c is a normalizing constant. I am trying to find the distribution of $G_1+G_2$. I have tried transforming ...
0
votes
0answers
18 views

How to approach a hypothesis test problem

I have a specific problem I'm working on, but I can simplify it to an example problem like this: I have two biased coins, one (A) which generates heads 10% of the time and one (B) which generates ...
1
vote
0answers
14 views

Convergence in distribution of distributions $p_n$ implies convergence in distribution of $s_n$?

Question Setup Suppose $p_n(x,y)$ is a sequence of probability densities on $\mathbb R^2$ and $q_n(x)$ is a sequence of densities on $\mathbb R$ such that \begin{align*} \int b(x,y) \ p_n(x,y) \ dx ...
0
votes
0answers
28 views

A Gaussian Divided by a Gaussian Equal to A Gaussian Divided by a Constant

I have a neural-network model in which each neuron is associated with an angle $\theta$. Firing rate as a function of $\theta$ is either a Gaussian or a constant. The claim has been made using this ...