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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1answer
11 views

Expected value of probability distribution

A plumber loads his truck each morning with faucets that will be needed for the service calls and other emergency calls that come in that day. Based on past experience, the number of faucets required ...
0
votes
0answers
9 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 ...
0
votes
0answers
13 views

Markov inequality help?

I'm trying to work through some problems and I've arrived at the following: For some random variable $T_{i}$: $E{T_{i}} \leq Cn^{2}$ with C some constant I want to show: $P(T_{i} \geq ...
0
votes
0answers
18 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
21 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 ...
-1
votes
0answers
14 views

Suppose that X1 and X2 denote a random sample of size 2 from a gamma distribution, Xi~GAM(2,1/2). Find the pdf of Y=sqrt(X1+X2) [on hold]

Suppose that X1 and X2 denote a random sample of size 2 from a gamma distribution, Xi~GAM(2,1/2). I must find the pdf of Y=sqrt(X1+X2. I substituted in the for theda and k into the distribution and I ...
0
votes
0answers
10 views

What is the Gini impurity index of an empty set?

Now, this may be a silly question because in practice you would never calculate the gini impurity on an empty set of observations. However, I did notice that while the shannon entropy is 1.0 for an ...
0
votes
1answer
20 views

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

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
0answers
11 views

Which model to be used for predictive analysis

I have a problem where i have been given set of data against month example Month | Data1 | Data2 1---------5--------5 2---------6--------7 Consider the data 1 be the temperature and data 2 be the ...
0
votes
1answer
13 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
14 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
16 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
vote
0answers
25 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
25 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
22 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 ...
-1
votes
0answers
12 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 ...
0
votes
1answer
17 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
vote
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 ...
0
votes
2answers
8 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
28 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
votes
0answers
9 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 ...
-1
votes
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.
0
votes
0answers
20 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
19 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
37 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
votes
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
votes
0answers
35 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} ...
0
votes
0answers
24 views

Probability question involving stochastic process [on hold]

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
216 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
33 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
votes
0answers
31 views

Markov Chain - Steady State behavior problem

I've been asked to solve the following problem. The problem: Let $X_n$ be a Markov chain with states in given space E, given transition matrix $P$ and all states belong to one and only recurrent ...
-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
0
votes
1answer
37 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
vote
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
39 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
20 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
38 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
61 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
23 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
42 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 ...
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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 <= ...
10
votes
6answers
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 ...