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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0
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1answer
224 views

Conditional expectation of second moment given sum of iid variables.

We have $\xi_i \geq 0$, $\forall i = \overline{1,n}$ (i.i.d. variables). Assume that $S_n = \xi_1 +...+ \xi_n$. It is easy to show that $\mathrm{E} (\xi_1\vert S_n = 1) = \frac{1}{n}$. Now we want ...
1
vote
0answers
19 views

CDF of maximum of iid rvs [duplicate]

I am having a small doubt regarding maximum of random variables. I have $$Z= \max\{ X_1, X_2,\dots X_p, \dots X_N\}$$ where all $X_i$ are independent, identically distributed. Now, If for sure, I know ...
1
vote
1answer
123 views

A fair dice is thrown six times and the list of numbers showing up is noted. The probability that among the numbers 1 to 6 only 4 nu…

Question : A fair dice is thrown six times and the list of numbers showing up is noted. Now how to find the probability that among the numbers 1 to 6 only 4 numbers appear in the list Please ...
2
votes
1answer
1k views

create a Gaussian distribution with a customize covariance in Matlab

the Matlab function 'randn' randomize a Gaussian distribution with $\mu= \begin {pmatrix} 0\\0\end{pmatrix}$ and $cov= \begin {pmatrix} 1&0\\0&1\end{pmatrix}$ Ineed to randomize a Gaussian ...
0
votes
2answers
47 views

There are eight males and 12 females in a certain club. In how many ways can a committee of five be chosen if it is to consist-

There are eight males and 12 females in a certain club. In how many ways can a committee of five be chosen if it is to consist Entirely of Males? Entirely of Females? 2 males and 3 females?
7
votes
1answer
127 views

Probability of an integer being a prime

$\Omega=\mathbb{N}^*,P(\omega=n)=\dfrac{1}{2^n}$, let $A_k$ be the event $k\mid\omega$. 1) Find $P(A_k)$ 2) Let B be the event "$\omega$ is prime", show that ...
1
vote
1answer
45 views

Given a probability distribution, how many times do I have to repeat an experiment so see a certain outcome

My question concerns random number generation under certain constraints. I assume that the random number generator is good enough to generate uniformly distributed numbers. This means that each number ...
1
vote
1answer
73 views

A generalization of the Glivenko-Cantelli theorem

Let $P$ and $P_n$ be probability measures on $\mathcal{B}(\mathbb{R})$ with distribution functions $F$ and $F_n$. Moreover, let $F$ be continuous and $(P_n)_{n\in\mathbb{N}}$ weakly converge to $P$. ...
2
votes
1answer
118 views

How to analyse a random walk with random transition probabilities

Consider a $1$-dimensional random walk with discrete time steps. We start at the origin and at each integer position there is possibly different probability of moving right one step, or left one step. ...
2
votes
1answer
80 views

Convex decomposition of a vector

Let $(a_i)_{i=1}^n$ be a probability vector, that is, $a_i\geq 0$ and $\sum_i a_i=1$ and let $(U_{ij})_{i,j=1}^n$ be a unistochastic matrix, that is, the pointwise square of a unitary matrix. Now ...
4
votes
1answer
118 views

Exercise from Norris' book on Markov chains

Let $(X_n)$ be a Markov chain on $\mathbb{N}$ with transition probabilities satisfying: $$p_{0,1}=1,\quad p_{i,i-1}+p_{i,i+1}=1,\quad p_{i,i+1}=\left(\frac{i+1}{i}\right)^{\alpha}p_{i,i-1}$$ The ...
1
vote
1answer
53 views

Sum of Two Poisson distributions

The probability distribution for the number of goals scored per match by Team A is believed to follow $X \sim Poi(0.8)$. Independently, the number of goals scored by Team B is believed to ...
1
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0answers
59 views

About the definition of mean square convergence.

A sequence of random variables $X_n$ is said to converge to $X$ in mean square if $$\mathbb{E}\left((X_n-X)^2\right) \rightarrow 0 \ \ \mathrm{as\ } n\rightarrow \infty$$. I understand what expected ...
1
vote
1answer
75 views

Simple Markov Chain: Random Walk on $\mathbb{Z}$

We are given a random walk on $\mathbb{Z}$, where $p_{i, i+1}= p < \frac{1}{2}$ and $p_{i,i-1}=1-p > \frac{1}{2}$, starting at $0$. Now we have to compute the probability that we eventually ...
0
votes
1answer
138 views

A random sample of size 5 is drawn from the pdf $f_{Y}(y) = 2y, 0\leq y \leq 1$. Calculate $P(Y_{(1)} < 0.6 < Y_{(5)})$.

A random sample of size 5 is drawn from the pdf $f_{Y}(y) = 2y, 0\leq y \leq 1$. Calculate $P(Y_{(1)} < 0.6 < Y_{(5)})$. (Hint: Consider the complement.) Attempt: The pdf of the largest order ...
0
votes
1answer
33 views

Expectation of a random variable that is similar to standard deviation distribution

Let's assume $\xi_i \sim N(0,\epsilon), i = 1,\dots, 9$ and $\xi_i$ are independent. How to compute next expectation? $$ E\sqrt{\frac{(\xi_1 - \frac{\xi_1 + \xi_2 + \xi_3}{3})^2 + (\xi_4 - \frac{\xi_4 ...
1
vote
1answer
33 views

Conditional expectation for random walks

The questions asks to $ E[X_1|S_n]$ where $ S_n = \sum_{[n]} X_i $ with $X_i$ i.i.d. of finite expectation. My attempt was to consider an arbitrary Borel set, pull it back under $ S_n $ to get a set ...
0
votes
1answer
90 views

Let $X$ and $Y$ have joint pdf $f(x,y)= 4e^{-2(x+y)}$; $0<x<\infty$, $0<y<\infty$. Find the CDF of $W=X+Y$

First I have to find the CDF of $W=X+Y$ which I tried to do this by substituting in the $w$ but it isn't working. Maybe I have the wrong bounds. Lastly I have to find the marginal pdf of $U$ which I ...
1
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0answers
75 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
votes
2answers
213 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), ...
0
votes
2answers
33 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 ...
0
votes
2answers
103 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
25 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 ...
2
votes
2answers
45 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 ...
1
vote
2answers
67 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
2answers
360 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 ...
1
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0answers
92 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ 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 $\mathsf kth$ moment of T about the origin, and state the condition for the ...
1
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2answers
129 views

Probability of the highest order statistic below the population median. [closed]

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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votes
1answer
28 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
votes
0answers
76 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
98 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
25 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
461 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 ...
2
votes
1answer
77 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 ...
1
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2answers
376 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 ...
2
votes
1answer
50 views

Basic probability and counting methods

A somewhat geeky problem has been on my mind the last few days: In my accommodation 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
59 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
81 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? ...
1
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0answers
41 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
188 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. ...
3
votes
0answers
62 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), ...
6
votes
3answers
330 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
vote
0answers
109 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
116 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)))?
-1
votes
2answers
164 views

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

Let X be an random variable on a given probability space and let 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 ...
2
votes
1answer
303 views

Almost sure convergence using Borel-Cantelli lemma

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 ...
11
votes
6answers
3k 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) = $\frac{1}{2}$. ...
2
votes
0answers
31 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 ...
2
votes
3answers
370 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. ...
2
votes
1answer
71 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, ...