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 (1)

0
votes
0answers
8 views

Joint distribution of arrival times in Poisson process

I need to compute the following joint distribution in a Poisson process: $f_{S_A S_{A+B}}(t_1, t_2), t_2\ge t_1$ $S_A$ and $S_{A+B}$ are the arrival epochs of the $A^{th}$ and ${A+B}^{th}$ arrivals ...
0
votes
0answers
7 views

Distribution of the ratio of two dependent chi-square

I look for my work the distribution of the ratio of two dependent chi-square variables $X, Y$ with different degrees of freedom for each one. Meanwhile I only found the distibution for the case where ...
2
votes
1answer
24 views

Uniform Distributions in Probability

X, Y, and Z are independent and uniformly distributed over [0,1]. I'm trying to find the distribution of XY by using the joint transformation T = X, W = XY. We haven't learned transformations yet, ...
0
votes
0answers
14 views

Determine if the following family of hash functions is universal

Let $H = \{h_1,h_2,h_3\}$ be the family of hash functions defined below, each mapping $\{a,b,c,d,e\}$ to $\{0,1,2\}$. Is $H$ universal? A family of hash functions is universal if $\forall ...
0
votes
1answer
8 views

Convergence in distribution of the negative part of centered/scaled poisson variable

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$ where $X_j ...
0
votes
0answers
10 views

Approximate normal distribution(this is different from what I asked earlier $\log(n)$ is replaced by $\sqrt{\log{n}}$)

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
0
votes
1answer
20 views

Determine the probability distribution of a ratio of two random variables?

Setting You are given two independent random variables $X_0,X_1$ with common exponential density $f(x) = \alpha e^{-\alpha x}$. Let $R = \frac{X_o}{X_1}$. Determine $\Pr[R > t]$ for $t > 0$. I ...
0
votes
0answers
6 views

Convergence of third moment in central limit theorem

Previously, I asked a question here about the rate of convergence of expectations of absolute values to the expected value of a Gaussian. If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = ...
2
votes
2answers
38 views

The limit of an expected value vs expected value of a limit in this betting game

Setting The outcome $X$ of a slot machine takes values 1,2,or 3 with probability $p(1) = \frac{1}{2}$, $p(2) = \frac{1}{4}$, $p(3) = \frac{1}{4}$. We are given 3 for one odds, that is if we bet 1 ...
0
votes
1answer
29 views

Central limit theorem in the setting of Poisson variables

Setting Given $S_{\lambda} \overset{d}{\sim} \operatorname{Poisson}(\lambda)$. Let $G_{\lambda}(t)$ be the distribution function of $\frac{S_{\lambda}}{\lambda}$. I need to determine ...
1
vote
1answer
19 views

What can you conclude about the first moment of a variable given the 3rd moment exists and is finite

Suppose you are given a random variable $X$ and told that $E[X^3]$ exists and finite. Can you conclude that $E[X]$ exists and is finite? What about $E[X^2]$? How would you argue rigorously whether ...
1
vote
1answer
12 views

Using chi-square test for statistics with multiple options in one variable

i think that this is more mathematical question, i am doing some statistics on survey. There is questions with multiple choices, so for example, if there are 4 choices, i can pick 1st ,2nd and 4th. ...
1
vote
1answer
22 views

Random Sample vs Simple Random Sample

I am reading, just for fun, the book Essentials of Statististics of Mario Triola. I am trying to see the differences between Random Sample and Simple Random Sample. In the book I found these ...
0
votes
0answers
15 views

Probability distribution for putting balls in boxes in a correlated way

I'm looking for help finding a probability distribution: Right now I have a problem where I have N indistinguishable balls, which I need to put into K indistinguishable boxes, each of which can hold ...
0
votes
2answers
21 views

Are these transient or recurrent states in a Markov chain?

I have the following transition matrix for a Markov chain with states $A, B, C, D, E$ $ \left| \begin{array}{ccc} 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & ...
7
votes
1answer
50 views

probability that no two spiders end up at the same vertex?

Eight spiders are located on the eight vertices of a cube. When a bell rings, each spider moves (at random, independent of the others) to an adjacent vertex. What is the probability that no two ...
1
vote
1answer
21 views

We have an urn with 5 blue balls and 15 red balls.

We remove 7 without replacement. Let R be the number of red balls removed and B the number of blue balls removed. Do you expect R and B to be positively correlated, negatively correlated, or ...
1
vote
0answers
25 views

Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
1
vote
1answer
30 views

Distribution problem where |a|, |b|, |c|, and |d| are at most 10. Check my work?

How many ways can a+b+c+d=18, where a,b,c,d are integers such that $|a|,\ |b|,\ |c|,\ |d|$ are each at most 10? This is what I have so far. If all four numbers have the restriction -10 =< a, b, ...
1
vote
1answer
32 views

Selecting n matches from two pockets.

Setting An eminent mathematician fuels a smoking habit by keeping matches in both trouser pockets. When impelled by need he reaches a hand into a randomly selected pocket and grubs about for a match. ...
0
votes
2answers
34 views

Total possible game scenarios

I am trying to figure out every possible scenario of every team in a league either winning losing or tying given the amount teams and weeks left in the season. For instance, the possible scenarios ...
0
votes
0answers
18 views

Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, ...
0
votes
1answer
22 views

Conditional distribution of geometric variables

Setting Suppose X1 and X2 are independent with the common geometric distribution w(k; p). Determine the conditional distribution of X1 given that X1 + X2 = n. Solution My argument is $$\Pr[X_1| ...
0
votes
0answers
29 views

Probability the pedestrian has to wait 3 time epochs to cross the street.

Setting A pedestrian can cross a street at epochs k = 0, 1, 2, . . . . The event that a car will be passing the crossing at any given epoch is described by a Bernoulli trial with success probability ...
-2
votes
2answers
41 views

Statistics and Probability (standard deviation)

Im finding this to be quite tricky, any ideas? A doctor is responsible for making treatment decisions for a group of patients who are suffering from a slow-acting non-fatal disease, x. The disease ...
6
votes
4answers
1k views

Probability that given a 1000 page book with 1000 misprints, a page will have 3 misprints.

Setting A book of 1000 pages contains 1000 misprints. Estimate the chances that a given page contains at least three misprints. Solution My solution is ...
-2
votes
3answers
39 views

Probability (independant events) [on hold]

Im sorry about this but the question doesn't seem to have enough info for me, could someone explain please. ...
0
votes
1answer
16 views

Ordering of elements drawn from uniform distribution

Setting $$X_1,\ldots,X_n \overset{iid}{\sim} \mathcal{U}[0,1]$$ Next order them so that $x_{(1)} \le x_{(2)} \ldots\le x_{(n)}$ Find $F_{(k)}(t) = \Pr[X_{(k)} \le t]$ in terms of a binomial sum, ...
0
votes
1answer
25 views

Expected earning when Player B randomly guesses a number player A picked

(Introduction to Probability, Blitzstein and Nwang) Player A chooses a random integer between 1 and 100, with probability pj of choosing j (for j = 1, 2, . . . , 100). Player B guesses the ...
1
vote
1answer
23 views

Assumptions of a probability distribution

Let $X$ be a continuous real-valued random variable indicating the fragility of a firm. Suppose that the firm defaults if $X$ takes a value above a threshold $u>0$. Hence $$ Prob(X>u) $$ is the ...
0
votes
1answer
16 views

How is it possible to write $\text {Pr} [M = m]$ where $M$ is random variable defined over a message space $\mathcal M$ and $m \in \mathcal M$.

In cryptography we consider random variables $K, M$ and $C$ over the key space $\mathcal K$ , message space $\mathcal M$ and cipher space $\mathcal C$, respectively. I've studied discrete mathematics ...
1
vote
0answers
12 views

How to determine the probability density function, ${f_{\dot X}}\left( {\dot x} \right)$, for the derivative process of a stochastic process?

I would like to calculate the up-crossing rate ($\nu _a^ + $) for a stationary stochastic process, $X(t)$, given by the probability distribution function of its 'intensity', ${f_X}\left( x \right)$, ...
0
votes
0answers
18 views

Synchronicity: probability of two related events happen in a while of time? [on hold]

"Synchronicity is the experience of two or more events as meaningfully related, where they are unlikely to be causally related. The subject sees it as a meaningful coincidence." Wikipedia What is the ...
-1
votes
0answers
25 views

Poisson Distribution for a rare event [on hold]

Suppose that commercial airplane accidents occur, on the average, twice in every 10^10 passenger miles (one passenger flying one mile). (Actually this nomenclature is slightly ridiculous and is meant ...
0
votes
1answer
23 views

Can this be a Markov Chain?

Suppose I have this game. In a bucket, I have $x$ number of balls, one of them is black. I randomly pick out balls out of the bucket one at a time (with no replacement). If I pick out a ball that is ...
0
votes
0answers
9 views

Functions of random variables - bivariate case

this is the question: I approached the first question in this way: Then, for the second question: After, my friend told me that if Z is a Poisson distribution than Var(Z) should be 25. I ...
0
votes
2answers
25 views

What is the average number of probes required to insert a new record into the hash table for this student?

Suppose a college has a master file consisting of $1000$ student records. Suppose that a hash table has been constructed to hold $3500$ records, and currently holds $1000$ records. A new student ...
0
votes
0answers
8 views

How write down PMF when random variable follows conditionally discrete uniform distributions with different support.

A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one of these ...
0
votes
0answers
12 views

Equiprobable spaces of infinite degree

Let us say that we randomly pick an integer from the set $Z$ , also as we know $|Z|=\infty$. Now if every element has an equal probability to get picked that probability $p$ must be $P = ...
1
vote
1answer
38 views

Algorithm for risky investments in banks

I made the following programming question on stack overflow but the users said it was more of math question. Here it is. Situation You start with a fixed amount of money, take it as $\$1000$. You ...
-4
votes
1answer
37 views

probability of randomness [on hold]

If you eat three apples, two squares, and seven artichokes, what is the probability that you will become green before you become seventy. I would like real thoughtful answers. Thanks in advance.
2
votes
1answer
26 views

Possibilities with unit digits and numbers

x is a three digit number greater than 700. If x is an odd number and each digit is not equal to zero, what is the possible number of x? (Replacement is not allowed) Answer: 91 Can somebody please ...
1
vote
1answer
24 views

What is the best choice given a probability and a cost for each choice?

I've been dealing with this problems for a few hours now and think I could use some outside help. The scenario is the following: We are given different choices with each one having a probability of ...
0
votes
2answers
83 views

How would I compute this sum?

So I would to compute this integral which is coupled by a sum: $$ \int_{x = 0}^{x = \lambda} \sum_{k=-\infty}^\infty e^{-( \frac{x-k \lambda}{\sigma} )^2} dx$$ I was thinking about using parseval's ...
1
vote
1answer
23 views

Given the density function: $\frac{1}{2}\exp\left(-\frac{x}{2}\right), \space x > 0$ find $P\left(\sum_{i=1}^{81}X_i > 170\right)$

Suppose that $X_1,X_2...X_{81}$ are independent random variable with the same probability density function $$\frac{1}{2}\exp\left(-\frac{x}{2}\right), \space x > 0$$ Find ...
0
votes
0answers
10 views

conditional expectation conditioned with XY [on hold]

1.Is the following right? $E[Z|XY]=E[Z|X,Y]$ what if $X$ is independent of $Y$? 2.If $X$ is independent of $Y$, is the following right? $E[X|XY]=X$
0
votes
1answer
31 views

Calculate the variance of $Y=2X+7$

Let $X$ have a mean of $4$ and variance of $4$. Let $Y=2X+7$. Calculate the variance of $Y$. I know that the formula for variance is just $E[(X-\mu)^2]$ so we would have $E[(X-4)^2] = E[X^2-8X+16] = ...
2
votes
1answer
49 views

Probability for having consecutive success in an experiment

A friend asked me the following question: "In an experiment, we are tossing a fair coin 200 times. We say that a coin flip was a success if it's heads. What is the chance for having at least 6 ...