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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4
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2answers
333 views

Let $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. Find $EZ$.

Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$. Now I need to find $EZ$. Should I start like ...
0
votes
0answers
101 views

How to find the cdf of the minimum of two r.v's?

Let $I=min\{0,W+V-U\}$ where $W,V,U$ are r.v's. Find the CDF of $I$ ?
1
vote
0answers
43 views

Second-order expected value given threshold

Suppose that $X_1, X_2, ..., X_n$ are uniform, independent random variables in $[0,1]$, and let $r\in[0,1]$. Suppose that the variables are ordered as $Y_1\geq Y_2\geq ...\geq Y_n$. Given that ...
2
votes
1answer
1k views

probability of hand with at least 2 kings

A hand H of 5 cards is chosen randomly from a standard deck of 52. Let E1 be the event that H has at least one King and let E2 be the event that H has at least 2 Kings. What is the conditional ...
1
vote
1answer
71 views

Binomial Random Variables: $P(X=k)$ as $k$ goes from $0$ to $n$

If $X$ is a binomial random variable with parameters $n$ and $p$, where $0 < p < 1$, show that As $k$ goes from $0$ to $n$, $P(X = k)$ first increases and then decreases, reaching ...
3
votes
2answers
203 views

biased coin flip. expected sequence

A biased coin $C$ has $\Pr(H) = a$ and $\Pr(T) = 1-a$. The coin $C$ is flipped $n$ times. What is the expected number of times that the consecutive sequence $HXH$ occurs where $X$ can be either $H$ or ...
0
votes
1answer
75 views

Joint distribution

This might be a trivial question. Let $X_{1}$ be a random variable and let $X_{2}$ be a random variable with the same probability distribution as the random variable $Y_{2}$. Question: Can I ...
2
votes
1answer
45 views

selecting marbles

An urn contains $r$ Red and $b$ Blue marbles. A fair coin is flipped. If the flip is Heads then $h$ Red marbles are added to the urn. If the flip is Tails then $t$ Blue marbles are add to the urn. Now ...
0
votes
1answer
42 views

Probability and Variance question

Can you help me solve this problem? Problem: You are playing a game, The game has 6 circles (O O O O O O) every time you play for 1 coin, each circle has a chance to be linked to the next one ...
4
votes
3answers
66 views

Verify my solutions to counting problems

I'm pretty sure I know the answers to these problems, but still want to double check. How many different ways are there of arranging all the letters of the string CALCULUSBOOK? Solution: $12!$ since ...
3
votes
2answers
56 views

Is the set $O:= \{(x,y) \in X \times X: d(x,y) < r \}$ Borel measurable?

Let $(X,d)$ a metric space and $\mathcal{X}$ its Borel sigma-algebra, i.e. the sigma-algebra generated by the open sets of $X$. Is the set $O:= \{(x,y) \in X \times X: d(x,y) < r \}$ $\mathcal{X} ...
3
votes
1answer
218 views

Selecting books from a shelf

A shelf contains 24 books. How many ways can 6 books be selected from these 24 with the restriction that no two selected books can be adjacent? So first we want to divide by 2 to fulfill the adjacent ...
2
votes
1answer
125 views

Probability of selecting jellybeans

8 red and 9 blue jellybeans are distributed randomly to 4 students. What is the probability that each student got at least one jellybean of each color? I am getting $\binom{7}{3} \binom{8}{3} / ...
1
vote
1answer
222 views

Expected number of coins taken by a pirate (problem with rounding)

We have N number of coins in a chest Two pirates are in a queue to take coins of the chest. When we draw some coins, the probabilities are all equal, so ${1,2,3....,k}$ all have the same probability ...
1
vote
1answer
700 views

Obscure Probability Question

Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean 104 and standard deviation 5 (information in the article “Mathematical Model of Chloride Concentration in ...
2
votes
0answers
65 views

If $x \sim U(Z_n^*)$ then $x^2 \pmod n\sim U(QR_n)$?

Define: $Z_n^*=\{x \in Z_n | \operatorname{gcd}(x,n)=1\}$ $QR_n=\{x \in Z_n | \exists r \in Z_n \; s.t. \; r^2 =x\}$ How can I show that $x \sim U(Z_n^*) \implies x^2 \pmod n \sim U(QR_n)$? Thank ...
4
votes
1answer
102 views

Double Jumps of a Poisson Process

If $N_t$ be a Poisson Process with rate $\lambda>0$, surely for any prescribed $t>0$, the probability that $N_t$ "jumps (at least) twice" at $t$ is zero, i.e. ...
0
votes
1answer
110 views

correlation between two different variables

I am studying stochastic processes and found the next problem: Let $A$ and $\Phi $ be two independent random variables such that $E(A) = 0$, $E(A^2) < \infty$, and $\Phi$ is uniformly distributed ...
0
votes
3answers
284 views

Is there an easy way to calculate the probability?

I wonder whether there is a way to calculate the probability of this problem. Mr Smith is a door-to-door salesman.For the last 30 days, he has been knocking at my door randomly trying to sell me ...
0
votes
1answer
20 views

Probability enquiry dices here?Is this correct?

I came across this problem in a book that I bought recently. Three dices are dropped. Find the probability that when we roll the dices: a) In all the dices we score 5. (Not as a sum) My answer: ...
2
votes
1answer
245 views

Covariance of Student's t-distribution

Another integral (this time it looks like a lot of work but maybe it can be simplified). I have the Student's t-distribution $$\int_{-\infty}^\infty ...
1
vote
1answer
253 views

Stochastic processes for beginers (good links and books)

I've a syllabus like that.. Markov chains with finite and countable state space. Classification of states. Limiting behavior of n state transition probabilities. Stationary distribution. Branching ...
4
votes
5answers
8k views

Probability of rolling three dice without getting a 6

I am having trouble understanding how you get $91/216$ as the answer to this question. say a die is rolled three times what is the probability that at least one roll is 6?
2
votes
2answers
138 views

Independence in Bernoulli scheme

We've got probability space $(\Omega, F,P)$ and Bernoulli scheme with $n$ trials and with success probability equal to $p$. $A_k$ means exactly $k$ successes in $n$ trials. Prove that for any $B \in ...
3
votes
1answer
56 views

Probability, rolling a dice

We roll a dice 10 times. What is a probability of obtaining a 6 in a first roll (event A) if we obtain 6 in all next 9 rolls (event B). Is it that simple that $P(A | B) = \frac{P(A \cap B)}{P(B)} = ...
0
votes
0answers
27 views

Second-Order Random Choice Proof

Given $G = (V,E)$ ;$ |V| = n, |E| = m$ then choose $T$ with $t$ vertices uniformly I have to proof the graph theory as $$E[X] - E[Y] \geq a$$ Which $$E[X] \geq \frac{(2m)^t}{n^{2t-1}}$$ X is random ...
4
votes
1answer
176 views

Expected Value of a Randomly decreasing function

We are asked to find the expected value of the following function RDF(N, K) for i = 1 to K do N = random(N) return N ...
1
vote
0answers
200 views

Drop $n$ random circles inside a square without overlap

The problem: find the size $l$ of a square such that $n$ circles with radius $r$ dropped randomly inside it will overlap with probability $t$. I'm writing a graph visualization tool, and would like ...
0
votes
3answers
167 views

Conditional expectation in the case of $\mathcal{A}=\{\emptyset,\Omega,A,A^c\}$

I have a small computation to do and I am not able to prove it: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probabiltiy space. Let $X$ be an integrable random variable and $A\in\mathcal{F}$ an event. ...
0
votes
1answer
200 views

Distribution of Product of Random Variables with one being the normal distribution.

Let X and Z be independent, with $X\sim N(0,1)$, and with $\textbf{P}(Z=1)=\textbf{P}(Z=-1)=\frac{1}{2}$. Let $Y=XZ$ (i.e., Y is the product of X and Z). (a) Prove that $Y\sim N(0,1)$. (b) Prove ...
-2
votes
1answer
59 views

Probability question math please?

a) We have $A_1$, $A_2$ and $A_3$ which are three events. Write the event: at most one event happens. Solution: I dont know :S b) We have $A$, $B$ and $C$, three independent events with ...
2
votes
1answer
211 views

An irregular 6 faced dice

An irregular 6 faced dice is such that the probability that it gives 3 even numbers in 5 throws is twice the probability that it gives 2 even numbers in 5 throws .How many sets of exactly 5 trails can ...
0
votes
0answers
27 views

Normal distribution interval

I am trying to find $c$ so that $P(|X - 5| < c) = .95$ with $\mu = 5, \sigma^2 = 4$. I came up with: $P( \frac{-c}{2} < Z < \frac{c}{2}) = .95$ Attempting to solve for $c$, the $z$-score ...
2
votes
0answers
998 views

Standardized Normal Distribution Problem

Mopeds (small motorcycles with an engine capacity below $50~cm^3$) are very popular in Europe because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the ...
4
votes
3answers
73 views

A probability question that involves $5$ dice

For five dice that are thrown, I am struggling to find the probability of one number showing exactly three times and a second number showing twice. For the one number showing exactly three times, the ...
1
vote
0answers
56 views

Showing sum diverges

I am working on a homework problem and I was wondering if someone could possibly give me a hint (or a solution) on how to show that the sum $\sum_{k\geq 1} (1- Q_k)$ diverges, where $$ Q_k = \sum_{0 ...
1
vote
0answers
43 views

Reference on the Open and Closed migration processes

I have read part of Frank Kelly's book on Open and Closed migration processes and I would like to find out a bit more. I am not an expert on the area with only some basic knowledge in continuous ...
9
votes
1answer
683 views

Expected area of the intersection of two circles

If we pick randomly two points inside a circle of radius $R$, and draw two circles centred at the two points with radius equal to the distance between them, what is the expected area of the ...
1
vote
1answer
51 views

An integral identity for unimodal distributions

For two unimodal densities $f_0$ and $f_1$ defined on $\mathbb{R}$, I figured out that $$\int_{-\infty}^{\infty}\sqrt{f_0(x)f_1(x)}\left[\ln(f_1(x))-\ln(f_0(x))\right]\mathrm{d}x=0.$$ Why should this ...
0
votes
1answer
271 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
0
votes
0answers
559 views

Hypothesis Testing on Exponential distributions

Let $X_1, \dots, X_n$ be independent exponential $(\theta)$ random variables. Suppose we are interested in testing $H_0: \theta = \theta_0 = 1$ versus $H_A: \theta = \theta_1>1$. Consider two tests ...
0
votes
0answers
572 views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
1
vote
1answer
52 views

Conditional Probability Mass function

How do I prove this probability equation: Suppose we have $N$ is Poisson($\lambda$), and if we have that $X$ is a binomial random variable Bin($N,p$). If we know what $N$ is we get: $$P(X=k| ...
1
vote
2answers
144 views

How to prove $\mathop{\rm Var}[X] \geq 0$ with Cauchy-Schwarz Inequality??

$\def\Var{\mathop{\rm Var}} \def\E{{\mathrm E}\,} $ X is non-negative r.v. Since, $\Var[X]$ is a sum of square so I knew that $$\Var[X] \geq 0$$ I have to prove that $\Var[X] \geq 0$ by using ...
1
vote
1answer
456 views

Confidence Intervals for an Exponential Distribution

$y_{1}$ is distributed $f_{Y}(y\mid\theta) = \theta e^{-\theta y} I_{(0, \infty)}(y)$, where $\theta > 0$. Analyze the confidence interval for $\frac{1}{\theta}$ given by $[L(Y), U(Y)] = [Y, 2Y]$. ...
14
votes
5answers
5k views

Probability that n points on a circle are in one semicircle

Choose n points randomly from a circle, how to calculate the probability that all the points are in one semicircle? Any hint is appreciated.
3
votes
1answer
122 views

What is the probability for a monkey writing Shakespeare? [duplicate]

The article clearly describes the idea but does not the state the probability. What is the probability? Can this problem be extended to audio, graphics and video for instance what is the probability ...
1
vote
1answer
794 views

Determining The Value, c, A Random Variable Assumes

The question I am working on is: In each case, determine the value of the constant c that makes the probability statement correct. $P(c \le |Z|)=0.016$ Here is my attempt: $P(|Z| \ge ...
2
votes
2answers
66 views

Let $F_X(x):=P(X\leq x)$ a distribution function of a random variable $X$. Prove that $F_X$ is right-continuous.

Let $F_X(x):=P(X\leq x)$ a distribution function of a random variable $X$. Prove that $F_X$ is right-continuous. I need to show that for every non-increasing sequence $x_n$ with $\lim x_n=x$ I will ...
1
vote
0answers
71 views

Survival Analysis

I have some survival times which are exponentially distributed for two groups G1 (treatment) and G2 (control). The data are censored with a censoring distribution given by h(c), so I only observe: 1) ...