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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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 ...
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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 = ...
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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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1answer
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1answer
39 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.
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1answer
31 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 ...
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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 ...
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2answers
91 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 ...
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1answer
24 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 ...
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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? $E[X|XY]=X$
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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] = ...
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2answers
64 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 ...
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2answers
10 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 ...
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1answer
44 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 $P(Y_m=1)<1$. ...
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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 ...
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2answers
25 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
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1answer
45 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 ...
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2answers
43 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 ...
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2answers
60 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 ...
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1answer
19 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       ...
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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 ...
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1answer
31 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 $W$.
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1answer
25 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 ...
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2answers
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2answers
40 views

Independence of two sequences of 1's and 0's.

Given a family $F_n$ that has $n \ge 2$ children, consider the two statements: A: $F_n$ has 0 or 1 girl B: $F_n$ has both boy(s) and girl(s). Also assume boys and girls are equally likely to be ...
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1answer
43 views

Check for independence of variables when the density (or distribution) is known.

This question is closely related to a previous one: Determine correlation and independence when only the joint density is given? Nonetheless, the setting is reproduced below. The joint pdf of $X = ...
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2answers
38 views

8th positive odd integer that is an ODD Catalan number? [on hold]

The $n^{\text{th}}$ Catalan number is given by the formula $C_n = \frac 1{n+1}\binom{2n}n$. It also satisfies the recurence \begin{align*}C_n &=\sum_{k=0}^{n-1}C_kC_{n-1-k}\\ &= ...
2
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1answer
36 views

At least 2 girls between every pair of boys distribution question?

Three boys and eight girls are seated randomly in a row of 11 chairs. All orders are equally probable. What is the probability that there are at least 2 girls between every pair of boys? What is ...
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0answers
12 views

Confidence level using unknown median

This is a general question. Lets say we are giving a sample space n. The sample space has an unknown median The order statistics of each person, $**X_{i}**$ in sample space is measured (lets say ...
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2answers
34 views

Expected value of a die when decision to role again depends on previous outcome

(Introduction to Probability, Blitzstein and Nwang) A fair die is rolled some number of times. You can choose whether to stop after 1, 2, or 3 rolls, and your decision can be based on the ...
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1answer
26 views

Calculating the probability of winning roulette after x bets

I'm going through all of my homeworks to study for my final and I'm getting hung up on this one problem I never figured out... A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you ...
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20 views

How to Justify the exclusion of some samples?

I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and ...
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0answers
12 views

CONFIDENCE LEVEL for Median Interval

A firm wants to estimate the unknown median, m , of the height of their employees. Random Simple Size = 90 $X_{i}$ is the order statistics of the Sample Size X where height of each employee was ...
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2answers
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Probability of 4 aces [on hold]

Hi guys if you can answer this it would be much appreciated. What is the probability of drawing all 4 aces on top of a deck of cards Thank you
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1answer
34 views

Determine correlation and independence when only the joint density is given?

The joint pdf of $X = (X_1,\ldots,X_n)$ is: $$f_{X}(x_1,\ldots,x_n)=\begin{cases} Ar^2,&0 \le r \le R\\[0.2cm] 0,& \text{ otherwise }\end{cases}$$ where $r = \sqrt{x_1^2 + \ldots + x_n^2}$ ...
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2answers
66 views

iPod Shuffle question

Suppose I had an iPod with x number of songs. Suppose one of them is my favourite song. I hit the shuffle button, so a random song plays. If it is not my favourite song, after the song ends, I will ...
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25 views

Express expected value with help generating function

I understand, how to express expected value with help generating function. For example, I have the following generating function: $D(z) = p K(z) + q M(z)$, where $p + q = 1$. How can I express ...
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11 views

detailed balance condition for coupled Langevin equation

Suppose $a$ and $m$ are real variables and they satisfy the following two coupled Langevin equations: $$ \dot{a}=F_a(a,m)+\eta_a(t);\quad\dot{m}=F_m(a,m)+\eta_m(t) $$ where $\eta_a$ and $\eta_m$ are ...
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0answers
12 views

Proving $\frac{d}{d\theta}\mathbb E\left[ \log\left( \frac{AY+BY+N}{ AY+BY \frac{X}{\theta^{-\alpha}} +N } \right) \right] \leq 0$

Let $X$ and $Y$ be exponentially distributed random variables with means $\theta^{-\alpha}$ and $(1-\theta)^{-\alpha}$, respectively. Simulation results suggest that $$\frac{d}{d\theta}\mathbb ...
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0answers
27 views

Existence of measure given the margin is a step function

Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function $$Q(x)=A, if \phantom{0}0\leq x < x^*$$ $$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$ s.t. $$A,B\in[0,1] ...
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0answers
51 views

Probability of $m$ out of $n$ rolls of a die being among the numbers $1,2,\ldots,m-1$, for some $m$.

Suppose I have a $k$ sided die with the numbers $1,2,\ldots,k$ on each side, and that I roll it $n$ times ($n<k$). What is the probability that there exists an $m\leq n$, so that $m$ of the $n$ ...
3
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2answers
67 views

If $X_i$ are iid $U(0,1)$ random variables, $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...
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0answers
54 views

Is it normal (correct) to calculate a probability without knowing the sample space?

Is it normal (correct) to calculate a probability without knowing the sample space? Background: I have finished a probability calculation $\mathbb{P}(E)$. I want to do some simulations. ...
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0answers
26 views

case deletion formula restricted least square estimator [on hold]

hi any one please help me to find out a case deletion formula for restricted least square estimator? $$ \hat\beta = (X' X)^{-1} X'y-(X' X)^{-1} R' [R' (X' X)^{-1} R]^{-1} R(X' X)^{-1} X'y $$ i need a ...
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1answer
18 views

Polynomial Chaos: How are the pdfs calculated from the response surface?

Lets assume one has the following response surface: $y(x,\xi) = \sum^N_{i=0} c_i H_i(\xi)$. Where $\xi$ is Gaussian and $H_i$ is the $i^{th}$ Hermite polynomial. I've seen a lot of papers show the PDF ...
2
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1answer
29 views

Find the unit vector so that this condition is true.

Let $(X_1,X_2)$ be jointly normal with density $$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$ Find unit vector ...
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1answer
16 views

Random Variable probability summation tweaking

I can't seem to figure out what they do to get to the bottom
5
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1answer
44 views

Definition of conditional probabiliy as function dependent on $\sigma$-Algebra

I know that for events $A,B$ with $P(B) > 0$ the conditional probability is defined as $$ P(A | B) = \frac{P(A \cap B)}{P(B)}. $$ Of course by regarding $A$ as constant, and varying $B$ we get a ...
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1answer
44 views

About the distribution of balls in bins

Suppose we have $n$ balls and $n$ bins, and consider the following process: at stage $k$, we throw $\ln{n}$ balls into the bins, independently at random. We stop after $n/\ln{n}$ stages, when all ...