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
40 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 ...
0
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0answers
5 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
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0answers
6 views

CDF of maximum of iid rvs

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 ...
3
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0answers
54 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 networks specifically when calculating coverage probabilitites- if needed I can provide reference- $$\mathbb{P}[ X \geq ...
1
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1answer
21 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
15 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
23 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 ...
6
votes
1answer
308 views

$m$ balls into $n$ urns

Assume that there are $m$ balls and $n$ urns with $m\gt n$. Each ball is thrown randomly and uniformly into urns. That is, each ball goes into each urn with probability $\dfrac1n$. What is the ...
1
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1answer
31 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 $P$. ...
7
votes
2answers
103 views

Roll a fair die until a 6 appears for the third time. What is the chance that all six values have occurred?

The question in the title is a homework question that I have been stumped on for some time. My approach thus far was to treat it as an occupancy problem. From class we derived the following formula ...
2
votes
2answers
26 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
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2answers
53 views

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

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 ...
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0answers
14 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
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1answer
27 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 ...
5
votes
1answer
60 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
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2answers
51 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
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2answers
17 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?
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0answers
32 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 ...
2
votes
1answer
27 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 ...
0
votes
1answer
19 views

Finding the density of rayleigh distribution

Suppose $T_{i}\sim$ iid $\operatorname{Ray} (\sigma^2=1)$ for $i= 1,2,...,n=20$. Find $P\left(\min(T_1,...,20)< t\right)$. Let $X=T_{(4)}$ and $Y=T_{(11)}$. Find $f_{(X,Y)}(x,y)$. I know that ...
0
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1answer
22 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?
1
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1answer
17 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
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1answer
31 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
3
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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
0answers
22 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. ...
0
votes
1answer
22 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
votes
0answers
24 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 ...
0
votes
2answers
54 views

Probability of Permutations/Combinations

How do you set up the formula for the probability of a permutation/combination? Question: If you have a group of candy with 2 Snickers, 4 Kit Kats, and 2 Butterfingers and you take two pieces out, ...
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votes
1answer
19 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 ...
0
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2answers
16 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 ...
1
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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 ...
0
votes
0answers
18 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 ...
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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
1answer
24 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 ...
1
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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 ...
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votes
0answers
15 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
1answer
18 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
0answers
14 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
18 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 ...
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0answers
40 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 ...
0
votes
1answer
30 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 ...
2
votes
3answers
225 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
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0answers
30 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
27 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), ...
1
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1answer
17 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 ...
-1
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2answers
16 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
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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: ...
0
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2answers
23 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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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 ...