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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statistical probablity

A man tosses two fair dice. What is the conditional probability that the sum of the two dice will be 7, give that (i) The sum is odd, (ii) the sum is greater than 6, (iii) the two dice had the same ...
2
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2answers
16 views

Pairwise independence vs independence

Two fair dice are thrown. We have three events: A: The first die shows an odd number B: The second die shows an even number C: Both are odd or both are ven Show that $A,B,C$ are ...
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0answers
10 views

Finding cdf, percentile, variance, and standard deviation from pdf.

$$f(x) = \begin{cases} 2(1-\frac{1}{x^2}) & \text{if }1\le x\le2 \\ 0 & \text{otherwise} \end{cases} $$ Compute the CDF of X: $$ \int^X_12(1-\frac{1}{y^2})dy = 2x+\frac{2}{X}-4 $$ So I ...
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0answers
11 views

Ehrenfest chain

In the Ehrenfest model,Let Xn denotes the number of balls in the left urn. And there are N balls total. When we calculate P(Xn+1=i+1|Xn=i, Xn-1=in-1,...,X0=i0), why don't we taking account of the ...
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0answers
8 views

Finding recurrent states given a Markov chain

I have trouble in approaching the problem where: Consider a Markov chain $X_n$ , $n ≥ 0$, with state space $S = N = {0, 1, ...}$ and transition function $$ p(x,y) = 1/7, y=0 $$ $$ p(x,y) = 2/7, y∈ ...
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1answer
26 views

I roll a fair die repeatedly until I get $6$, what is the probability that neither $1$ nor $2$ occurs before $6$ appears.

I roll fair a die repeatedly until I get $6$, what is the probability that neither $1$ nor $2$ occurs before $6$ appears. Not sure how to go about this.
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0answers
14 views

Poisson Counting Insurancee example [on hold]

An insurance company finds that for a certain group of insured driver , the number of accidents over each 24 hours period rises from midnight to noon and then declines until the following ...
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3answers
23 views

Probability of an even number of sixes

We throw a fair die $n$ times, show that the probability that there are an even number of sixes is $\frac{1}{2}[1+(\frac{2}{3})^n]$. For the purpose of this question, 0 is even. I tried doing ...
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2answers
26 views

What is the probability that when a deck of cards is shuffled and dealt, exactly 3 of the 4 aces will be dealt within the last 20 cards?

I am trying to figure out this problem, I think that it is a "permutations with repetition" type of question.
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1answer
17 views

Probability: Finding the Number of Apples Given Two Scenarios

You have a bag containing 20 apples, 10 oranges, and an unknown number of pears. If the probability that you select 2 apples and 2 oranges is equal to the probability that you select 1 apple, 1 ...
1
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1answer
10 views

Probability density function in Rayleigh distribution

It says that $$ f(x;\theta) = (x/\theta)e^{-x^2/(2\theta^2)}, x>0 $$ is the Rayleigh distribution. And asks to verify that $f(x;\theta)$ is a legitimate pdf. Can you explain how to verify ...
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1answer
27 views

Find $c=c(n)$ so $T = c \sum_{i=1}^{n} |X_{i}|$ is an unbiased estimator.

I'm having some trouble trying to solve the following problem: Assuming that $X =(X_{1},\ldots,X_{n})$ is a random sample from the normal distribution with mean $0$ and unknown standard deviation ...
0
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1answer
20 views

Moments of a random sum with bounds Poisson distributed?

We have that $N$ and ${X_1,X_2,\dots}$ are all independent and that $f(x)=Cx^2(1-x)^2$. Then, we have: $$Z=\sum_{j=1}^{N+1}X_j$$ $N$~Poisson$\lambda$. Find the expectation and the variance of $Z$. ...
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2answers
32 views

Does the sum of Poisson random variables have a Poisson distribution?

So I have been taught that the sum of Poisson random variables have a passion distribution. However, I have a problem with this. Suppose you have a Poisson random variable $X$ with $E(X) = a$. Then ...
0
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1answer
21 views

Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers. [on hold]

Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers $Y_1, Y_2, ...$ from the uniform distribution on $[0, 1]$, until ...
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1answer
9 views

Pseudo-inverse of the Cumulative Distribution Function of X

The goal of these calculations is to write a Python function that generates pseudo-random values with the distribution described below. This isn't relevant to the question (or even to this ...
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0answers
12 views

Borel isomorphism between polish spaces

In my lecture on stochastics the following result has been used: For any uncountable Polish space $X$ there is a Borel isomorphism between this space and the real line. I was not able to find a ...
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2answers
30 views

How does one find the mode of a distribution without counting manually?

I know if I have a set of elements $\lbrace 1,2,3,4,4,4,5,8,9\rbrace$ Then the mode is $4$ in this case. How do I find the mode for more complex distributions? I have formulas that give me ...
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1answer
36 views

Conditional Probability Problem: Two Radios from Two Factories

Q: There are two local factories that produce radios. Each radio produced at factory A is defective with probability .05, whereas each one produced at factory B is defective with probability .01. ...
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0answers
24 views

Probability of at most $K$ consecutive zeroes in a sequence of 0s and 1s.

I am studying for a probability in computer science course and came upon this exercise problem that I have trouble solving. I need this to prove that in a sequence W of length n, consisting of 1s and ...
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1answer
24 views

Use Maximum Likelihood Estimation to guess which dice got selected

We have two six-sided dice (faces numbered 1 through 6) and two tetrahedral dice (faces numbered 1 through 4). Someone selects two of them and throws each once. Then they tell us the sum of the ...
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1answer
19 views

Urn with white and black balls, random variable, conditional probability

An urn contains white and black balls with $p_w=p$ and $p_b=1−p$. If I made some extractions with replacement, what are the support and the probability function of $X_a$, where $X_a$ is the random ...
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1answer
20 views

Calculate characteristic function

$p(n)=(1-r)^2nr^{n-1},n=1,2,...$ $f(z)=1/(1-z)$ has derivative $f'(z)$ with convergent power series $f'(z)=1/(1-z)^2=1+2z+3z^2+...$ the answer I have got is $(1-r)^2e^{it}(1-re^{it})^{-2}$ , I am not ...
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4answers
61 views

$2$ players take turns and draw from a box containing $1000$ balls, $3$ of them are black.

I'm not sure how to tackle this question. Assume a box containing $1000$ balls, $3$ of them are black and the rest are white. $2$ players $A_1$ & $A_2$ take turns and draw from the box without ...
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0answers
17 views

Application of Doob's optional stopping theorem to an elementary probability problem

The elementary probability problem is as follows. Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables such that $X_k \sim U(0,1)$ for each $k$. Define $\tau := \inf\{n\geq 0: ...
2
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2answers
31 views

Average number of events happening if each happens with $p=\frac{1}{n}$ and we run it $10000 n$ times.

Let an event $e$ have probability of happening $\frac{1}{n}$. Let us assume we have $m$ independent possibilities for similar events to happen. With $m>>n$. What is the average number of times ...
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0answers
28 views

Probability Mass Function of a Sentence

We have a sentence: Some dogs are brown. We choose one letter (out of the 16) at random. Let Y be the length of the whole word containing the letter. How can I find the probability mass function of Y? ...
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1answer
48 views

How did they calculate the possible endings? [on hold]

On this link @edit you can see all the possibilities of endings. The game has six stages, on each you have 3 choices and at the end, you have 5 stages with 2 endings each. Its like: 1. > 2a 2b 2c > ...
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0answers
22 views

The “how many pieces do you have buy on average” problem, a markov problem?

I recently discovered a problem similar to this one in a book about Markov chains: Assume you can buy $n-$ different set of cards in a store, but you do not know which one you'll buy: What is the ...
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1answer
25 views

Given probability distribution $f(x)=2-bx$ find $b$ and range for $x$

Suppose that the distances between houses and the center of a city are distributed with the density function: $f(x)=2-bx$, where $x$ denotes distance. If this is a proper density function, what can we ...
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2answers
57 views

Given any 40 people, at least four of them were born in the same month of the year [on hold]

Given any 40 people, at least four of them were born in the same month of the year. Why is this true?
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1answer
22 views

Calculate the characteristic function $\varphi_W$ of W

$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise. I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.
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1answer
29 views

Let $E(X)=\mu$ and $\operatorname{Var}(X)=\sigma^2$. If $E(Y|X)=a+bX$, find $E(XY)$ as a function of $\mu$ and $\sigma$.

I can't figure out the answer for a question on my econometrics course. Somehow it seems simple, but still I can't seem to figure it out. Maybe I am thinking the wrong way about it. Could someone ...
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1answer
23 views

Conditional entropy and independent conditioning variables

Let $X,Y,Z,Y',Z'$ be random variables where $Y\sim Y', Z\sim Z'$, $Y$ and $Z$ are independent, while $Y'$ and $Z'$ are, in the sense that we have $p(X,Y,Z)=p(X|Y,Z)p(Y)p(Z)$ ...
2
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1answer
39 views

Find a continuous PDF on $[0,6]$ for given probabilities

Find a continuous probability density function $f$ on $[0,6]$, such that $\mathbb{P}([0,2]) = 0.6$, $\mathbb{P}([1,4]) = 0.5$ and $\mathbb{P}([3,5]) = 0.2$. After some calculations I came up with ...
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0answers
34 views

If $P(X_1 < X_2)$, what is $P(X_1 < X_2 \cap X_1 < X_3)$?

Say $X_i$ can have a real value in the range [1,100]. All $X_i$ are independent of each other and all values are equally likely. So then $\mathbb{P}(X_1 < X_2) = \frac{1}{2}$, right? But then, ...
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0answers
25 views

Poisson Process with continuous rate, Finding Conditional Number of Arrivals

Poisson with customer arrival to the shop rate given by $\lambda (t)=16-(t-4)^2$ Calculate $P(N(5)-N(3)=40|N(4)=70)$ where $N(i)$ means the number of arrivals in the first $i$ hours. The shop ...
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1answer
14 views

Probability of a vector of normal distribution

Given the set of vectors $\{\mathbf{g}^{1}, \ldots, \mathbf{g}^{N-1} \}$ where $\mathbf{g}^{i} \in \mathbf{R}^M$. Assume that $N \leq M$ and elements of $\mathbf{g}^{i}$ follows normal distribution, ...
1
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3answers
47 views

Find the distribution of $Y = -\log (1-X)$ given that $X\sim U(0,1)$.

If $X \sim U (0,1)$ then if we define a new random variable $Y=-\log (1-X)$ then what will be distribution of $Y$. Please explain.
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0answers
24 views

Appropriate distribution for set of probabilities $p_1 ,…, p_n$

I am doing some evaluation of a system, that has set of probabilities $p_i$ $i= \in \{1,...,N\}$, I need to model them as random variables such that : $$ \sum_i p_i \leq 1$$ and $$ 0 \leq p_i \leq 1 ...
2
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0answers
27 views

Showing a relation between binomial and negative binomial analytically

If $X$ is binomial random variable $B(n,p)$ and Y is negative binomial $(r,p)$, How can I show that $F_X(r-1) = 1- F_Y(n-r)$. While it is possible to show that using the definition of binomial and ...
1
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1answer
29 views

Derive the distribution of $Z$ given two identically and independently exponentially r.v.s?

$$Z=\frac{X}{X+Y}$$ $(X,Y)$ are iid r.v.s with $$f(x)=\lambda e^{-\lambda x}$$ We are asked to condition on $Y$ to derive the distribution of $Z$; $F(t)$ and $f_Z(z)$. I don't know where to ...
2
votes
1answer
23 views

What is $\operatorname{Pr}\{X_j=0|X_i=k\}$ [on hold]

Suppose $u_n=\operatorname{Pr}\{X_n=0|X_0=1\}$ What is $\operatorname{Pr}\{X_j=0|X_i=k\}$, where $\{X_n\}$ is a branching process and $k\geq 0$, if we were to write the answer in terms of the ...
3
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0answers
28 views

Markov Chain: Steady State Distribution.

A total of $M$ balls are divided between two urns A and B. A ball is chosen uniformly at random. If it is chosen from urn A then it is placed in urn B with probability $b$ and otherwise it is returned ...
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1answer
16 views

If $P$ is a transition matrix, and $m_{ij}$ the mean return time, how to show $m_{ij} = 1+ \sum_{k \neq j}P_{ik}m_{kj}$?

If $P$ is a transition probability matrix of a finite state regular Markov Chain, and $m_{ij}$ is the mean return time, how can I show that $m_{ij} = 1+ \sum_{k \neq j}P_{ik}m_{kj}$? It seems rather ...
-1
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0answers
32 views

show that $Y_1$ is unbiased for $\theta$ and find its variance [on hold]

Let $X_1,\ldots,X_n \stackrel {\text{iid}} {\sim} \text{$P_0$}(θ)$ $$Y_1= \frac {X_1+3X_2+5X_5} {9} $$ $$ Y_2= \sum_{i} X_i$$ Show that $Y_1$ is unbiased for $\theta$ and find its variance. Show ...
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1answer
27 views

Expected value problem: flip $6$ fair coins before we obtain $3$ heads and $3$ tails?

How many times on average (expected value) must we flip $6$ fair coins before we obtain $3$ heads and $3$ tails? I know I need $∑ xp(x)$. I just don't know how to apply it.
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1answer
23 views

Why does $E(C\cdot \epsilon\; \vert\; C\cdot X) = E(C\cdot \epsilon\; \vert\; X)$?

Let $C$ be an $n\times n$ matrix, $X$ is $n \times k$, $\epsilon$ is $n \times 1$ This is taken from a simply proof of strict exogeneity in an Econometrics textbook by Hayashi. The explanation he ...
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0answers
8 views

When does a periodic but positive recurrent markov chain have a limiting distribution

So I know it's a fact that an aperiodic, finite state, irreducible (so positive recurrent) markov chain has a unique stationary distribution which is limiting. However, I am curious if there is a ...
2
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2answers
38 views

What is the probability that a psychic correctly “predicts” the outcome of at least 5 out of 10 coin flips?

Assume the psychic is actually just randomly guessing on each flip. The attempt: let E be the event in question number of outcomes per flip = 2 chance of correctly guessing the correct outcome = ...