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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0answers
15 views

Let $X_n>0$ be iid and $P(X_n>t)\sim t^{-\alpha}$, show that $Y_n=n^{-1/\alpha}S_n$ and $1/Y_n$ are tight.

We are given that $X_n>0$ be iid with common distribtuon $X$, and $P(X>t)\sim t^{-\alpha}$, I need to show that the scale of $Y_n$ is $n^{1/\alpha}$. Or in other words show that ...
0
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2answers
24 views

Probability of scratch and win card

A game of “scratch-and-win” is played as follows. You scratch 2 out of 3 covered circular tabs on a game coupon • • • to reveal 2 images. The coupons are of types (A), (B), (C) with images ♥ (heart), ...
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2answers
20 views

Probability of an event happening while another doesn't

Say you have a bag with $5$ numbers $(1,2,3,4,5)$. What is the probability that I will draw a $1$ if I draw $3$ times (no replacement)? What is the probability that I will draw a $1$ if I draw 3 ...
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0answers
10 views

to find face values of a biased die rolled n times assuming the probabilities [on hold]

Write a MATLAB function that would simulate rolling a biased (“lucky”) die N times; i.e., the function must return face values of N rolls of a biased die (N is a function input). Assume that the ...
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1answer
16 views

Given a pdf $f_{Y}(y)$ and $n$ random observations. Find probability that last observation will be the smallest number in all the sample?

Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the entire sample? attempt: ...
1
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1answer
13 views

Determining a conditional probability with a random variable.

Assume $X$ is a normal distributed random variable with mean $2$ and variance $4$. Determine the conditional probability $P(1 \le X \le 3|0 \le X \le 4)$ What I did: $$Z_0 = \frac{0-2}{2}=-1$$ $$Z_1 ...
0
votes
1answer
20 views

Convergence in distribution - Proof

I was given a problem: For each $n\in\mathbb N$, let $X_n$ be a random variable with uniform distribution over the set $\{0,\frac{1}{n},\frac{2}{n},\dotsc,\frac{n-1}{n},1\}$. Let ...
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0answers
2 views

Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of ...
0
votes
1answer
26 views

Let X be the amount won or lost in betting $5 on red in roulette.

HW Problem here, not sure where I'm messing up. Let $X$ be the amount won or lost in betting \$5 on red in roulette. Then $P(5) = \frac{18}{38}$ and $P(-5) = \frac{20}{38}$. If a gambler bets on red ...
0
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0answers
7 views

Kth moment of the standard deviation from a normal population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the Kth moment of T about the origin, and state the condition for the existence ...
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2answers
12 views

Probability of the highest order statistic below the population median.

What is the probability that the highest order statistic of a random sample of size n from any continuous distribution is below the median ( population median ) of that distribution.
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0answers
18 views

Presentation of 2 images in a random but counterbalanced way

Problem: For 18 trials randomly a ‘left’ labeled image or ‘right’ labeled image is shown. The first 9 trials should contain the opposite number of left images as the last 9 (a.k.a. counterbalance). ...
-2
votes
1answer
18 views

Probability: How much days we need to play a game win

Suppose the probability of win a lotery game is : $1/1000$ If a person play the lotery every day with the same combination, how much time he need to wait to win the lotery? Im thinking to use a ...
4
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0answers
36 views

How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
-1
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0answers
6 views

local time process and markov process [on hold]

Is the local time process of an semimartingale a Markov process? If not, under what conditions, the local time process of an semimartingale becomes a Markov process?
2
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0answers
33 views

Does this strategy look correct to you (solving for probability density function with three Random Variables)

The following formula is a formula I got from a paper that deals with wireless network (if needed I can provide reference) $$\mathbb{P}[ X \geq T( Y+Z )] = \int_{-\infty}^{\infty} ...
1
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0answers
21 views

Probability question involving stochastic process

A stochastic process $\{x_{k}\mid k=1,2,3,...\}$ of zeroes and ones is given with the property that $x_1 = 1, x_2 = 0$ and for every $k>2$ it is true that the probability of the event $x_k = 1$ is ...
1
vote
1answer
7 views

Interval of probabilities which satisfy a Markov chain

Given the following markov chain, where T1 is the start state, the labels are shown on the state( 'a' in this case) and p and 1-p are probabilities for that transition happening: Now, for what ...
2
votes
3answers
208 views

Probability of no ace in a 6 card hand, given 4 are not aces.

A player is dealt six cards out of a normal deck of cards. He looks at the first four and notices there is no ace among them. What is the probability that he does not have an ace at all. This sounds ...
1
vote
1answer
31 views

Find the chance that $a^3 + b^3 \equiv 0 (\mod 3)$

We are given set of integer numbers $\{1,2, \dots N\}$. $N \ge 3$ Then perform a drawing with replacement of two elements $a$ and $b$. Problem is to find the probability of following statement holding ...
-2
votes
1answer
18 views

number of possible outcomes in a license plate with conditions [on hold]

howmany license plates can me made when a) first two letters are different and the rest different digits e.g. DA3457 b) two letters in alphabetical order and the digits increasing e.g. CD1234
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votes
1answer
32 views

Prove that if $X$ is stochastically larger than $Y$ then $E(X)\ge E(Y)$

Prove that if $X$ is stochastically larger than $Y$ (i.e. $P(X > t) \ge P(Y > t)$ then $E(X)\ge E(Y)$.I understand how to solve the problem if $X$ and $Y$ are non-negative random ...
1
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1answer
30 views

Basic probability and counting methods

A somewhat geeky problem has been on my mind the last few days: In my accomodation at Uppsala there are 12 rooms to a floor. I discovered the other day that another British girl whom I know lives ...
2
votes
1answer
35 views

A problem on distributing indistinguishable balls into 10 different groups such that…

I got this problem which I am stuck at for an hour and half: Suppose that we have an infinite number of indistinguishable balls and we need to distribute them into 10 different groups such that $ ...
-1
votes
1answer
28 views

Let A be the set of irrational numbers in [0,1]. Show that P(A)=1

Let A be the set of irrational numbers in [0,1]. Show that P(A)=1 , where P is Lebesgue measure. What ever we do there are infinite irrational numbers for every two rational numbers, right? and we ...
1
vote
0answers
17 views

Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
0
votes
1answer
14 views

What is a probability ensemble?

The definition I have says An ensemble index by I is a sequences of random variables indexed by I. Namely, any X = {X_i}_{i \in I}, where each X_i is a random variable, is an ensemble indexed by I. ...
2
votes
0answers
36 views

Does this non-negative non-increasing function eventually attain $0$

Let $\phi(z): \mathbb{R}\rightarrow [0,B]$, with $B>0$, be a non-negative and non-increasing function such that $\phi(0) = B$ and \begin{align} \phi(z) = \max(0, E[\phi(z+X)]+a\mu - c), ...
5
votes
3answers
60 views

Secret Santa Perfect Loop problem

(n) people put their name in a hat. Each person picks a name out of the hat to buy a gift for. If a person picks out themselves they put the name back into the hat. If the last person can only ...
1
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0answers
22 views

A property of the hazard function of the normal distribution

I have a problem that I can't figure out. Define $\Gamma\left(x\right):=\frac{\phi(x)}{1-\Phi(x)}$, where $\phi(x)$, $\Phi(x)$ are the density respectively cumulative distribution function of the ...
0
votes
2answers
38 views

Expected value of the sum of the two largest values from a Uniform parent

Is the expected value of the sum of two greatest values in an uniform distribution in [0,1] of n random variables (x1,x2,x3,x4,...,xn) equal to E(max(x^n))+E(max(x^(n-1)))?
0
votes
2answers
42 views

Show that Y=aX+b is an random variable.

Let X be an random variable on a given probability space and lrt a,b∈R. Show that Y=aX+b is an random variable. if X has a distribution function F, what is the distribution function of Y? if X ...
1
vote
1answer
24 views

calculating probability [on hold]

There are $3$ boxes. In each boxes a random number is added. The number can be from $0$ to $255$. Let's say we added, randomly, these numbers: $5\,\,\,\,\,\,25\,\,\,\,\,\,199$ What is the ...
1
vote
1answer
22 views

Almost Surely convergence using Borell Cantelli

Let $X_n$ be a sequence of random variables. I want to show that if $E[X_n] \rightarrow C$ and $Var(X_n) \leq \frac{C}{n^2}$, where $C$ is some constant, then $X_n$ converge almost surely to $C$. I ...
-1
votes
0answers
17 views

CLT, mle, variance [on hold]

This is a practice problem that I don't know how to do. Let X_1,...,X_n be an i.i.d. sample from an exponential distribution with the density function. f(x/T) = (1/τ)*e^(-x/τ) 0<= x <= ...
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4answers
2k views

Probability that last child is a boy

Johnny has 4 children. It is known that he has more daughters than sons. Find the probability that the last child is a boy. I let A be the event that the last child is a boy, P(A) = $\frac{1}{2}$. ...
0
votes
0answers
14 views

Probability of X streaks of length Y in total coin flips Z

It's a simple problem to state but I'm having a hard time finding an answer. I would like to know what the probability is of finding X streaks of length Y in total coin flips Z. Any help would be ...
-1
votes
0answers
13 views

Finding the density of rayleigh distribution

Suppose $T_{i}$~ iid $Ray (\sigma^2=1)$ for $i= 1,2,...,n=20$ -Find $P(min(T_1,...,20)$< $t$). -Let $X=T_{(4)}$ and $Y=T_{(11)}$. Find $f_{(X,Y)}(x,y)$ I know that the density of the Rayleigh ...
2
votes
0answers
11 views

Estimate of shared variance for n samples of x and y

I am performing a t-test on n different samples of both $X_1, X_2,...,X_k$ and $Y_1,Y_2,...,Y_k$. To begin with I want to assume that all 2*n samples have the same variance but that they do not have ...
0
votes
0answers
17 views

Properties of Identically Distributed RVs.

I've a little doubt in part (iii) of the question posted above First I wrote the PMF of Z \begin{vmatrix} Z = X+Y & -2 & -1& 0 & 1 & 2\\ P(Z=z) & .09 & 0.24 & 0.34 ...
2
votes
3answers
26 views

Probability of obtaining a double six in at least two throws

The question: A pair of fair dice is thrown 10 times. What is the probability of obtaining a double six in at least two throws? My attempt: Let X denote the total number of double sixes obtained. ...
-3
votes
1answer
39 views

Probability that a monkey at a type writer types “hamlet” [duplicate]

A monkey types each of the 26 letters of the alphabet exactly one time. What is the probability that the world "hamlet" appears somewhere in the string of letters?
1
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1answer
24 views

What is the probability of not rolling any given number on 10 rolls of a die?

In other words, ALL combinations which don't contain at least one of the number from 1-6 would count. So for example... 5, 2, 3, 3, 4, 1, 5, 5, 3, 1 would be counted because there is no 6 Also 5, ...
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votes
2answers
22 views

Probability of being Second Highest

Suppose there are $n-1$ draws from a uniform distribution $[0, 1]$, then I get a draw from the distribution. What is the probability that if I shout out an $x$, then $x$ will be exactly the second ...
1
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0answers
21 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(x=0)$.Here is my solution... $\mu= \frac {p(2) 2!}{p(1)}$. then how can find the mean..thanks
0
votes
1answer
16 views

Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$

Let $U$ have a uniform distribution on $[0,1]$. Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$ My attempt: $F_Y(x)=P[Y\le x]=P[{1\over ...
1
vote
1answer
20 views

Probability of Sample Variance Given Variance

I am trying to solve a problem that I have never seen before and cant seem to find a way to solve it so any help or tips would be appreciated! Here's the Problem: Suppose a considerable amount of ...
0
votes
1answer
26 views

Equivalence of $\sigma$-algebras: generated by $[a,b]$ and $[-\infty,b]$

Show that the $\sigma$-algebras generated by the collection of all intervals of the form $[a,b]\subset\Bbb R$ and by the collection of all the intervals of the form $[-\infty,b]\subset\Bbb R$ are ...
1
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0answers
23 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
0
votes
1answer
33 views

Extended Bayes' theorem: p(A | B, C, D)

I'm having some difficulty understanding Bayes' theorem with multiple events. I'm trying to put together a Bayesian network. I have four independent probabilities but I have found that A, B and C can ...