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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1
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2answers
19 views

Probability of an event based on percentage in fixed lapse of time.

I am a software engineer. I am also a former triathlete that rides with a large group of friends every time we have a chance. i am trying to come up with a little software to distribute among us ...
0
votes
0answers
10 views

Estimate ratio of two expectations by sample means

I have a question about the estimation of a ratio of two expectations. Suppose $X_{i}$ and $Y_{i}$ are two random variables with $i=1,\cdots,N$. We seek to estimate $\mathbb{E}X_{i}/\mathbb{E}Y_{i}$ ...
0
votes
0answers
33 views

Probability - coin flip with known tails count and probability of heads is known.

First coin toss out of $10$ will always be tails. Five tails will appear in total (including the first tail) out of $10$ flips. The probability of heads is $0.3$. What is the probability for exactly ...
2
votes
1answer
55 views

Number of inversions

Compute the sum of the number of inversions that appear in the elements of $S_n$. In other words find the total number of inversions that the elements of $S_n$ have combined. I mean how can we ...
0
votes
0answers
20 views

Average number of $5$-card draws before all $52$ cards in a deck are drawn.

So an interesting question was brought to me today and I'm not sure how to formulate the equation to answer it. A person draws $5$ cards from a deck, writes the cards down, puts the cards back in the ...
0
votes
1answer
28 views

Is this a special probability distribution?

Does the distribution function: $\frac{1}{\theta}e^\frac{-y}{\theta} $ Have a special name? If not, how can I find the variance? I keep running into a dead end when I try.
1
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0answers
18 views

Normal Approximation to the Binomial

I need to solve this problem using Normal Approximation to the Binomial Distribution to check if the value is similar to the one that I found using the Binomial distribution. Question: What is the ...
0
votes
1answer
33 views

calculating probability using mean and standard deviation [on hold]

The time that it takes to assemble a piece of machinery is well modeled by the normal distribution with mean of 72.9 minutes and standard deviation of 8.55 minutes. What is the probability that it ...
0
votes
1answer
28 views

Binomial distribution births

I am trying to solve this problem using binomial distribution. What is the probability that in a group of $100$ people, $10$ of them were born in either March and April. Binomial distribution ...
1
vote
2answers
35 views

Probability of the next number in a random sequence being the largest seen so far

Suppose I have a uniform random number generator producing a sequence of random numbers in the range $0...100$. I am trying to work out what the probability is that the $n^{th}$ number in this ...
-2
votes
0answers
20 views

Probability of independent event, Slotting machine model [on hold]

A slot machine has four separate wheels that rotate independently. On each wheel are four pictures of a lemon and one picture of a cherry. Each time the slot machine level is pulled, one picture on ...
2
votes
2answers
29 views

What is the probability that nobody receives the same ranking twice?

Four players compete in a tournament and are ranked $1$ to $4$. They then compete in another tournament and are again ranked from $1$ to $4$. Suppose that their performances in the second ...
0
votes
0answers
32 views

How to solve the Probability Markov chain system of equations

I have this system of equations from a 2-D Markov chain (see the figure. How can i calculate the coefficient matrix, state probability vector and the constant vector from this system of equations. ...
-2
votes
2answers
26 views

If $\mathbf{E}(e|x) = 0$, then $\mathbf{E}(h(x)e) = 0$ for any function $h(x)$ [on hold]

Consider a random scalar $e$ and a random vector $x$. Let $\mathbf{E}(e|x) = 0$. Show that $\mathbf{E}(h(x)e) = 0$ for any function $h(x)$ I am asked to show this, but I have no clue where to ...
0
votes
0answers
15 views

Moments of censored exponential distribution

I have a question as to whether my calculation of moments of censored exponential distribution is correct. I have two random variables $T_A=\min(\tau,t_1)$ and $T_B=\min(\tau,t_2)$, where $t_1<t_2$ ...
0
votes
2answers
29 views

The skewness coefficient for given pdf?

$$f(x)= \begin{cases} 0.5-\frac{x}{8},& 0\le x \le 4 \\ 0, & \text{otherwise} \end{cases} $$ I have found $E(X) = \frac{4}{3}$ and $Var(X) = \frac{8}{9}$ Th problem says that the skewness ...
1
vote
1answer
21 views

Expectation of scaled sum of squares of iid random variables

Let $X_1, \dots, X_n$ be iid standard normal random variables. Consider the vector $X = (X_1, \dots, X_n)$ and the vector $Y = \frac{1}{\|X\|}(X_1, \dots, X_k)$ for $k < n$. What is ...
-4
votes
0answers
31 views

statistical probablity [on hold]

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 ...
3
votes
2answers
32 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 ...
2
votes
1answer
30 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 ...
0
votes
1answer
20 views

Ehrenfest chain

In the Ehrenfest model, let $X_n$ denotes the number of balls in the left urn. And there are $N$ balls total. When we calculate $P(X_{n+1}=i+1|X_n=i, X_{n-1}=i_{n-1},...,X_0=i_0)$, why don't we take ...
0
votes
0answers
11 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∈ ...
1
vote
1answer
38 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.
-1
votes
0answers
15 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 ...
2
votes
3answers
29 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 ...
-1
votes
2answers
39 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? [on hold]

I am trying to figure out this problem, I think that it is a "permutations with repetition" type of question.
3
votes
1answer
27 views

Probability: Finding the Number of Pears 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
vote
1answer
16 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 ...
0
votes
1answer
31 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
votes
1answer
25 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 $Z$. ...
1
vote
2answers
35 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 ...
1
vote
1answer
42 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 ...
0
votes
1answer
25 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 ...
1
vote
0answers
17 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 ...
0
votes
2answers
37 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 ...
0
votes
1answer
75 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. ...
0
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0answers
29 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 ...
0
votes
1answer
27 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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
22 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 ...
1
vote
4answers
68 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 ...
3
votes
0answers
24 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
votes
2answers
42 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 ...
0
votes
0answers
30 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? ...
-3
votes
1answer
49 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 > ...
1
vote
0answers
27 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 ...
-1
votes
1answer
27 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 ...
-5
votes
2answers
76 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?
0
votes
1answer
23 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.
3
votes
1answer
37 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 ...