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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3
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0answers
26 views

Shooting bullets

This is from http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/May2014.html Every second, a gun shoots a bullet in the same direction at a random constant speed between 0 and 1. The ...
0
votes
3answers
35 views

Is it possible for a reality to exist where the law of large numbers does not apply?

Being more specific, is the law of large numbers more empirical than it is rational? That is, is it more a feature of the observable universe that it is something that is true based on our definition ...
1
vote
2answers
12 views

Calculating the variance of speed measurements

Some speed measurements (km/h) outside Furutåskolan has been observed. They are supposed to be outcomes from a random variable with expectation . Result: $29, 31, 36, 34, 33$ (a) Construct a ...
-1
votes
0answers
7 views

Show the sample mean $\mathfrak{T}_t$ converges to the population mean faster than $n^{1/3}$.

Let $\mathfrak{T}_{t}$ be an iid random variable with support $\mathfrak{T}_{t} \in [0,1]$. Prove $n^{1/3}\frac{1}{n} \sum\limits_{t=1}^{n} (\mathfrak{T}_{t} - \mathbb{E}[\mathfrak{T}_{t}] ) ...
1
vote
1answer
7 views

Combined Distribution of Random variable

How to compute $P[T1 \le T2 \le t]$ for T1, T2 is independent random variable with exponential distribution in terms of cmf, pdf of T1 and T2? Similarly for $P[T1 \le T2 \le T3.. \le t]$ ? I tried ...
3
votes
0answers
19 views

Find probability that random triangle covers centre of circumscribed circle

We are given the equilateral triangle A. On each edge of the triangle we pick a point: randomly (probability distribution is Gaussian) independently of others We construct new triangle B ...
4
votes
2answers
23 views

Showing that $p^n(1-p) \leq \frac{1}{en}$

I am reading a paper and found the following Lemma without a proof. Let $X_1, \ldots, X_{n+1}$ be independent Bernoulli random variables, where $\Pr[X_i = 1] = p$. Let $E$ be the event that the first ...
0
votes
2answers
10 views

Covariance of uniform distribution and it's square

I have $X$ ~ $U(-1,1)$ and $Y = X^2$ random variables, I need to calculate their covariance. My calculations are: $$ Cov(X,Y) = Cov(X,X^2) = E((X-E(X))(X^2-E(X^2))) = E(X X^2) = E(X^3) = 0 $$ because ...
0
votes
2answers
21 views

Find the probability density function of $Y=X^2$

Consider the random variable X with probability density function $$f(x)=3x^2$$ if $0<x<1$, and $$f(x)=0$$ otherwise. Find the probability density function of $Y=X^2$. This is the first question ...
0
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0answers
4 views

Reference requests for an opt-cited result in Jennrich (1969)

Lemma 2 on page 637 of Jennrich (1967) states that: Let $Q$ be a real-valued function on $\Theta\times Y$ where $\Theta$ is a compact subset of a Euclidean space and $Y$ is a measurable space. ...
4
votes
3answers
42 views

Probability that a cow is black given that I've observed at least one side is black

I'm on a farm with six cows; three are white, two are black and one is completely black on one side and completely white on the other. I see one cow from the side, who appears to be black (that is, ...
2
votes
2answers
25 views

Probability a truck full of stones weighs more than 1800kg?

Stones of a particular kind weigh in mean 10 kg and have standard deviation 1 kg. Assume the weight of a truck is normal distributed with mean 1000 kg and standard deviation 100 kg. 50 stones are put ...
0
votes
1answer
16 views

expected value of three uncorrelated random variables

Random variables ξ, η and ζ are pairwise uncorrelated. It means that E(ξ*ζ) = E(ξ)*E(ζ), etc. Is it true that in this case E(ξηζ) = EξEηEζ ? How it can be proven? Note: we don't know if they are ...
6
votes
4answers
1k views

% of % - Please Help Me Prove My Friend Wrong

Here is the situation: My friend and I are at an impasse. I believe I'm correct, but he's so damn stubborn he won't believe me. Also, I'm not the most articulate at explaining things. Hopefully ...
0
votes
1answer
25 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 ...
0
votes
0answers
14 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
0answers
22 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 ...
0
votes
2answers
19 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?
5
votes
1answer
70 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
19 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
35 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 $P$. ...
0
votes
1answer
26 views

In how many distinct ways can a group of letters be ordered? [on hold]

In how many distinct ways can the letters aaabbbbb and aaabbbbbcccc be ordered?
2
votes
0answers
25 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
32 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 ...
0
votes
0answers
27 views

Probability - Runners in a race [on hold]

Consider a race with N runners, where N is unknown. Each runner is assigned at random a unique number between 1 and N. Suppose a group of n runners is observed crossing the finish line. Let z denote ...
3
votes
1answer
27 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 ...
-2
votes
1answer
25 views

Expected value of probability distribution [on hold]

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 ...
1
vote
1answer
24 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
1answer
17 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 ...
1
vote
0answers
21 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
25 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
18 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
16 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
33 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
19 views

Which model to be used for predictive analysis [on hold]

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
22 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
15 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
19 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
36 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
29 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
24 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
16 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
19 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
18 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
29 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
20 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
32 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
vote
0answers
35 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
votes
2answers
19 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
37 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). ...