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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6 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
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0answers
11 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 ...
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0answers
11 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
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2answers
17 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
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1answer
42 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
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1answer
20 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 ...
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0answers
10 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
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1answer
27 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
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1answer
31 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. ...
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2answers
33 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
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0answers
12 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
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1answer
17 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
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0answers
28 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 ...
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2answers
35 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 ...
5
votes
4answers
590 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 ...
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3answers
32 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
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1answer
15 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
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1answer
23 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
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1answer
21 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
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1answer
15 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 ...
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0answers
11 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', ${f_X}\left( x \right)$, ...
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0answers
17 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 ...
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0answers
22 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
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1answer
20 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
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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
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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
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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
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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
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1answer
35 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 ...
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votes
1answer
33 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
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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
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2answers
79 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
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1answer
22 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
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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
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1answer
30 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
48 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 ...
0
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2answers
9 views

Showing that $n1_{ \lbrace U<1/n \rbrace}$ converges to $0$ almost surely

Let $U \sim \text{Uniform}[0, 1]$ and $X_n = n1_{\lbrace U< 1/n \rbrace}$. I want to show that $X_n$ converges to $0$ almost surely. My attempt: I use Fatou's Lemma with the reasoning that if I ...
5
votes
1answer
43 views

prove this martingale inequality

The problem is like this: Let $Y_1,Y_2,\ldots$ be nonnegative i.i.d. random variables with $E(Y_m)=1$. Let $X_n=\prod_{m\leq n} Y_m$, show that $\lim_{n\rightarrow \infty}X_n=0$ if $P(Y_m=1)<1$. ...
1
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2answers
33 views

I can't understand this difference equation step

I am working on birth-death processes and I can't understand a step that is taken in a proof. The mean of a process is defined as $$\mu(t) = \sum_{n=1}^{\infty}np_n(t)$$ At certain stage in the ...
1
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2answers
24 views

Determine the expected value of a geometric distribution given some generic underlying distribution.

This is a variation of the standard waiting time problem. Suppose you have a sequence of variables $$X_0,X_1,X_2,\ldots \overset{iid}{\sim} F(x)$$ where $F(x)$ is continuous. And random variable ...
2
votes
1answer
40 views

Prove or disprove convergence in distribution of a poisson variable.

Let $$S \overset{d}{\sim} Poisson(\lambda).$$ I would like to determine $\frac{S-\lambda}{\sqrt{\lambda}}$ converges in distribution as $\lambda \rightarrow \infty.$ So my set up is: $$\Pr\left[a ...
1
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2answers
41 views

Probability of ultimate extinction? Need to show that an infinite series is less than $1$

I have the following probability generating function for a branching process - $$G_n(s) = \frac{n}{n+1} + \sum_{r=1}^{\infty}\frac{n^{r-1}}{(n+1)^{r+1}}s^r$$ It says in a book that extinction is ...
0
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2answers
56 views

Probability that n people collectively occupy all 365 birthdays

The problem is quite simple to formulate. If you have a large group of people (n > 365), and their birthdays are uniformly distributed over the year (365 days), what's the probability that every day ...
0
votes
1answer
18 views

Given an unfilled pmf, How to compute the Correlation coefficient?

This is the format in which I was given the PMF. Sorry for the messy table, don't know how else to make a table. Given this pmf $x$$y$ $f_{xy}(x,y)$ 1       ...
-1
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1answer
38 views

Can you prove the Law of Large Numbers?

So clearly it is not hard to experimentally prove that the more times something is done, say rolling a die, the closer your experimental results come to your theoretical likelihoods, but is there a ...
-1
votes
1answer
29 views

Find expected value of $W$, when $ W $ is the largest of the variables. [on hold]

Let $X_1, X_2,\ldots, X_8$ be independent exponential random variables of mean $1/2$, Let $W$ be the largest of the $X_1, X_2, \ldots, X_8$. Compute the expected value of $W$.
2
votes
1answer
24 views

Does Binomial variables independence implies Bernoulli variables independence

$X$, $Y$ are independent variables with Binomial distribution. $X={\sum_{i=1}^nX_i}$, $Y={\sum_{i=1}^nY_i}$. $X_i$, ($1\le i\le n$) are independent Bernoulli variables. Same applies for $Y_i$ Is the ...