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
9 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
31 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

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?
1
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0answers
21 views

Solving the following probability density function

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} ...
0
votes
0answers
15 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
119 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
17 views

number of possible outcomes in a license plate with conditions

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
0
votes
1answer
25 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 ...
0
votes
0answers
24 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
27 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
35 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
59 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
20 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
36 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 ...
0
votes
0answers
16 views

CLT, mle, variance

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 <= ...
10
votes
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
13 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 ...
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votes
0answers
12 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
10 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
9 views

Transient random walks on $\mathbb{Z}$ with step $\pm a_n$ of probability $2^{-n}$

Construct a transient random walks on $\mathbb{Z}$ as follows: For $n>0$, the step is $\pm a_n$ (to be chosen) with probability $2^{-n}$ . What I thought was that $a_n$ has to grow exponentially. ...
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
37 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
vote
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, ...
0
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
vote
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 ...
0
votes
1answer
16 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
vote
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 ...
0
votes
0answers
8 views

Upper bound for graphs with no k-cliques

We know that for random graphs $G(n,p)$ we have: $P[X=0]\leq e^{-\Theta(E[X])}$ where $X$ denotes the number of k-cliques in the random graph. Can this fact be used to say anything about the number of ...
3
votes
2answers
42 views

What is the Laplace transform of this random variable?

Define a random variable that takes only one value for example $$X=c$$ where c is a positive constant. What does the Laplace of it evaluate to i.e the following $$\mathcal{L}_X(s)= ...
0
votes
1answer
19 views

Probability mass function for the number of defective light bulbs among selected

This is the problem I have: There are $10$ light bulbs in a carton and $3$ of them are defective. Suppose $2$ light bulbs are randomly selected without replacement and let $X$ be the random variable, ...
1
vote
2answers
29 views

Variance of two functions

I have a problem where Var(X) is given as 8100, Var(Y) is given as 10,000. Var(X+Y) = 20,000. If X is increased by 500, Y is increased by 8%, such that the new formula is X+500 +(1.08)Y. How would I ...
0
votes
1answer
12 views

AP Probability problem on independence

This is a in-class practice problem. Suppose that the probability that a person has to park illegally and that he gets a parking ticket is 0.07. Last year Sam recorded data and found that because of ...
0
votes
1answer
35 views

Shortcut to finding $E(XY)$

The question says "Find $E(Y|X)$ and hence evaluate $E(Y)$ and $E(XY)$" The joint pdf is $$f_{X,Y}(x,y)=\begin{cases} 8xy, & \text{ for } 0< y< x < 1, \\0, & \text{ elsewhere } ...
0
votes
1answer
14 views

Number of graphs with M edges that does not contain K-clique

If we consider the space of graphs $G(n,M)$ with $n$ vertices and $M$ denotes the number of edges. Is there any way of upper bounding the number of graphs in this space that does not contain any ...
1
vote
1answer
33 views

Monotone Class Theorem Application

I am trying to proof the following statement. Let $h$ be a bounded, $\mathbb{F}$-predictable process with $\tau$ a $\mathbb{H}$-stopping time, we then like to prove \begin{equation} ...
1
vote
1answer
16 views

Joint Probability distribution for $Z=X/(X+Y)$

Suppose X and Y are two independent random variables with exponential distributions Exp(1) $Z=X/(X+Y)$ Find $P(Z<z)$ and show the random variable Z has uniform distribution. Thanks
2
votes
2answers
56 views

What is the probability that a number chosen at random in $[0,1]$ is transcendental?

Consider the interval $[0,1]$.What is the probability that a number chosen at random in $[0,1]$ is transcendental? Please give me some points on how to start this problem.Thanks
2
votes
0answers
15 views

Distribution of the square of magnitude of a Nakagami random variable [on hold]

Given the random variable $$h \sim \operatorname {Nakagami} (m,1)$$ $$ f_{h}(h)= \frac{2m^m}{\Gamma(m)} h^{2m-1} \text{exp}(-m h^2)$$ What is the distribution of the following function $$g:=|h|^2$$