Tagged Questions

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
votes
4answers
12 views

$12$ people $p_1, .. , p_{12}$ divided into 3 groups, what is the probability that $p_i$ and $p_j$ be in the same group?

Given the following question: $12$ people randomly divided into $3$ groups $g_1, g_2, g_3$, $g_1$ have 3 members , $g_2$ have 4 members, $g_3$ have 5 members. Each person $p_n$ belongs ...
0
votes
0answers
14 views

Bayes spam filtering

To analyze the words that appear in spam emails, you collect a sample of 1000 emails marked as spam and 1000 emails marked as non-spam. Of the 1000 spam emails, 210 contained the phrase This isn't ...
3
votes
1answer
21 views

Distribution of functions of uniform random variables

Given these two independent and uniform distributed random variables, $$X \sim U[-\pi,\pi]$$ and $$Y \sim U[-\pi,\pi]$$ What is the distribution of $$\sin(X)$$ and $$\sin (Y)$$ and the distribution ...
2
votes
0answers
11 views

Does convergence in total variation to convergence of moments?

Suppose $d_\mathrm{TV}(\mathbb{P}_n,\mathcal{N}(0,1)) \to 0$ for a sequence of probability distributions $\mathbb{P}_n$. Is each $k^{th}$ moment of $\mathbb{P}_n$ necessarily finite after some $n_k$? ...
1
vote
2answers
15 views

weak convergence or distribution! [on hold]

Let $(X_{n})_{n\geq 1}$ a sequence of random variables, the density of $X_{n}$ given by $$f_{X_{n}}(x)=\dfrac{nx^{n-1}}{\theta^{n}}, \ 0<x<\theta$$ Prove $X_{n}\rightarrow X$ distribution (or ...
0
votes
0answers
12 views

Is the square of the sum of Nakagami random variables the same as the sum of Nakagami RV squared?

Having random variable $U_i$ Nakagami distributed with parameter m, for $i\in[1:n]$. What would be the distribution of the following two examples be $$ \big|\sum_i {a_i U_i}\big|^2$$ where $a_i$ is ...
1
vote
1answer
28 views

Harder than usual deck of cards probability problem

Charlie draws five cards out of a deck of 52. If he gets at least three cards of one suit, he discards the cards not of that suit, and draws as many cards as he discarded. What is the probability he ...
0
votes
3answers
29 views

Throw a dice 4 times. What is the probability `6` be up at-least one time?

First time I approach a probability question (: Throw a dice 4 times. What is the probability 6 be up at-least one time? Intuitively, I would say: ...
0
votes
0answers
7 views

Probability of traveling from one city to another with probability p of a mudslide blocking roads

Two roads join Ashville and Benson, and two further roads join Benson to Carlyle. Ashville is directly connected to Carlyle using a railroad. All four roads and the railroad are independently blocked ...
-2
votes
3answers
28 views

Four Coins Are Tossed [on hold]

If four coins are tossed, what is the probability that exactly three of them show heads or exactly three of them show tails?
2
votes
1answer
29 views

Find the expected distance travelled

A photon starts in a random direction (uniformly distributed) from the center of a square (all of whose sides are mirrors) and hits a side (and reflects off of it). What is the expected value of the ...
0
votes
1answer
18 views

Almost a Frechet distribution but not quite yet

I have function as $$\frac{2}{\alpha}x^{\frac{2}{\alpha}-1}e ^{-x^{\frac{1}{\alpha}}}$$ This kind of reminds me of the Weibull and Frechet distribution but not quite because if it were I should be ...
0
votes
0answers
38 views

understand probability theory profoundly

I have probability and statistics course this semester. our text book is http://www.amazon.com/Probability-Random-Variables-Stochastic-Processes/dp/0071226613 and ...
0
votes
0answers
19 views

Finding the number of non empty urns after 9 steps

I'm trying to understand this example given in the book and am having trouble. The example states. Suppose that balls are successively distributed among 8 urns, with each ball being equally likely to ...
1
vote
3answers
19 views

Conditional probability with liars

Suppose an urn has $n$ balls of different colors and only one of them is white. Let there be two independent witnesses who speak the truth each with probability $.1$. A ball is selected at random from ...
0
votes
1answer
17 views

What is the joint distribution of sample mean and sample variance of normal distribution?

${X_i} \sim N\left( {\mu ,{\sigma ^2}} \right)$, define $\overline X =\dfrac{1}{{n}} \sum\limits_{i = 1}^n {{X_i}} $, ${S^2} = \dfrac{1}{{n - 1}}\sum\limits_{n = 1}^n {{{\left( {{X_i} - \overline X} ...
0
votes
0answers
11 views

Probability of finding roots to a multivariate polynomial [on hold]

Let there exist a $n$-variate $1$-degree polynomial ring. Suppose there is a finite set $S$ as a subset of the field over which the polynomial ring is defined. What is the probability that elements ...
0
votes
1answer
15 views

Probability questions based on mutual exclusion

I'm appearing for an exam for which I'm giving mock tests, however I came across this particular question that I'm unable to solve, it says: A certain experiment has three possible outcomes. The ...
2
votes
2answers
123 views

Probability (X >Y) when X and Y have the same distribution?

This is a problem from HW4 Joe Blitzstein's Harvard Stat 110 course. Let X be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so X takes values 1, 2, . . . , 7, with equal ...
0
votes
0answers
14 views

Is there of list of probability distributions ranked by “tightness of fit” for a given SD?

Given the mean and standard deviation of a distribution, we know from Chebyshev's theorem that at least a half of the values will be between between ($\mu - \sqrt{2}\sigma, \mu + \sqrt{2}\sigma$). If ...
2
votes
1answer
30 views

Probability of $B$ winning a series of games

$A$ and $B$ are two players. The probability of $A$ winning a particular game against $B$ is $1/3$ and the probability of $B$ winning the game is $2/3$. They play a series in which the rules are ...
0
votes
0answers
14 views

Definition and implications of Infimum

Let $\theta$ be a parameter. Let $X_i$ be a random variable. Let $F$ the joint probability distribution of data $\{X_i\}_{i=1}^{n}$. Let $(\theta,F)\in \mathcal{F}$. Let ...
1
vote
1answer
23 views

How to find the $E(N)$ using $E(M)$ where the $M$ and $N$ follow slightly different scenarios

An author sends his first manuscript to a large number of publishers, $C, D, E, ...$ , in turn, only approaching each one, after the first, if the one before has refused it. There is a constant ...
0
votes
1answer
23 views

Probability of intersections of independent events with a twist

I'm trying to solve this problem: Let $(A_n)$ be a sequence of independent events. Show that if $I$, $J$, are (finite/countable) disjoint sets, then $$ {\mathbb P}{\Large[}~\bigcap_{i \in I}A_i ...
0
votes
1answer
5 views

Distribution under null-hypothesis and type 1 error

Given random variables $X_1,...,X_n \overset{i.i.d.}{\sim} N(\mu, \sigma^2)$ where the variance $\sigma^2$ is known let the null hypothesis be $H_0: \mu = \mu_0$ For the statistic $T=\sum_{i=0}^nX_i$ ...
0
votes
1answer
16 views

Interpretation of Standard Deviation independent of the distribution?

Is there any intuitive way to interpret the standard deviation regardless of the probability distribution? So for example, for the normal distribution, I know how to interpret being within 1 standard ...
1
vote
1answer
13 views

Discrete Bivariate Distributions, find constant $c$

I am given $f(x,y) = c(x + y)$, and I have to find constant $c$ such that $f(x,y)$ satisfies the conditions of being a joint pmf for two discrete random variables $X$ and $Y$. $x = 1$, 2, 3, and $y ...
0
votes
0answers
30 views

Find two random variables with same distribution…

I am looking for two random variables and a probability measure $P$, such that $X$ and $Y$ have the same distribution, $X$ and $Y$ are dependent and $X$ and $Y$ are uncorrelated. I tried to take the ...
1
vote
0answers
22 views

Computing a Finite Expectation

Assume $1\leq\ k<m<n$ are positive integers and $X_1,X_2,...X_n$ are i.i.d. Geometric($p$) random variables. For all $j\geq\ k$ define $I_j=[(i_1,i_2,...,i_k):1\leq\ ...
1
vote
2answers
22 views

Solving for N in a binomial distribution

Mid-term study... Two dice are rolled. How many times must the dice be rolled so that the probability of getting a sum of 10 or greater on at least one roll is larger than 0.9? So am I correct ...
3
votes
3answers
41 views

Card Game Probability 13 Card Hand

Me and my friends play a four person poker style card game. Each person is dealt 13 cards, and it is a standard trump card game. Now, as the standard, a five card flush beats a five card straight, but ...
0
votes
1answer
20 views

proof of weak convergence of probablity measure [duplicate]

Let $(\mu_n)_{n\in\Bbb{N}}$ be a sequence of probability measure on $\Bbb{R}$ with characteristic functions $(\phi_n)_{n\in\Bbb{N}}$. Assume that $\lim_{n\rightarrow\infty}\phi_n(t)=1$ for all ...
0
votes
0answers
8 views

Sum of Lomax random variables

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. ...
1
vote
0answers
5 views

Conceptual Question on Cramer rao lower bound for performance measure

In system identification, parameter estimation I have found in several papers that an analytical bound is derived which is the CRB of the error variance of the estimates. For, optimal performance of ...
0
votes
1answer
24 views

Probability from multiple trials

This questions is from a practice mid-term that I don't have a solution to. A monkey in a research lab is given 6 tiles with the letters AAABNN. On each trial the monkey randomly arranges the ...
1
vote
1answer
15 views

A hand of six cards is dealt from a standard poker deck. Find formula for p_(XYZ) (x,y,z).

A hand of six cards is dealt from a standard poker deck. Let X denote the number of aces, Y the number of kings, and Z the number of queens. a) write a formula for p_(XYZ) (x,y,z). b) Find ...
-3
votes
2answers
19 views

If 4 balls are drawn without replacement, what is the probability that at least 3 black balls are drawn? [on hold]

There are 9 black balls and 10 red balls in an urn. If 4 balls are drawn without replacement, what is the probability that at least 3 black balls are drawn?
0
votes
1answer
55 views

lottery game probability

In the "Make Money Game," the winning number is four digits, each selected at random from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, e.g. 0-3-9-6, 0-0-6-0, 9-4-7-9. A player may place any of the following types ...
0
votes
0answers
20 views

Lack of memory of a geometric distribution, proving a general case.

I have to prove this for a general value so $P(X > j+k | X>j) = P(X > k)$ Using the conditional probability I get that $P(X > j+k | X>j) = \dfrac{P(X > j+k) \wedge P(X > ...
2
votes
2answers
29 views

Frankie and Johnny game. What should Johnny strategy if he wants to minimize his expected loss?

Frankie and Johnny play the following game. Frankie selects a number at random from the interval $[a, b]$. Johnny, not knowing Frankie’s number, is to pick a second number from that same inverval and ...
0
votes
0answers
14 views

MTG: Probability of drawing a card with enough mana to play it (part 2)

This is a continuation of the original question: Let's assume we have the same scenario 12 White 13 Black 3 Spell Card 32 Other cards Based on the answers in Part 1, I can now answer that ...
4
votes
2answers
84 views

Probability brainteaser

I find this braintease on the internet and do not know how to solve it. For the second question, My first thought is to deduct from the situation when there is only 2 slots, then 3, 4, .., n,.. slots. ...
1
vote
1answer
18 views

What is the probability of success?

If I have 12 Possible questions, of which 5 are asked and I only need to answer 2 of them, what is the probability of my success (i.e., I am able to answer 2 of the 5 asked questions) if I learn 2 of ...
1
vote
1answer
28 views

Deck of Cards Probability Question - Probability of Getting At Least 2 Queens

There was actually another question like this but the final answer a person mentioned was incorrect and I was confused as to how he got it. Can any answers explain how they got there? I'd like to ...
2
votes
3answers
25 views

Symmetric Distribution of Random Variable

Prove: Let $X$ and $Y$ be random variables with the same distribution. If $X$ and $Y$ take only two values​​, then $X - Y$ are symmetrically distributed around zero. Note: 1 - You can use ...
2
votes
2answers
15 views

Probability of an event happening

Studying for a mid-term with a practice test, and there's no solution, so I want to make sure I have this right. A fire alarm has the property that it will ring with 99.5% probability, if there is ...
1
vote
1answer
12 views

Distribution of random variable $Y$ passed throught distributin function of $X$

If \begin{align*} F(x)=P[X \le x] \end{align*} is continuous in $x$, show that $Y=F(X)$ is measurable and that $Y$ has uniform distribution \begin{align*} P[Y \le y]=y, \, 0 \le y \le 1. ...
0
votes
1answer
23 views

Expected value of a sum of random events

Suppose there's a market that has decided to award its most loyal customers. The market sells a certain type of breakfast cereals with a single token in each box. There are n different types of ...
1
vote
3answers
17 views

Standard deviation…

I have this random variable $X = \{-1, 0, 1\}$ with uniform repartition $p(X = -1) = p(X = 0) = p(X = 1) = \frac{1}{3}$. Expected value is $$E[X] = \sum_{i\in\{-1,0,1\}} x_ip_i = 0$$ Then variance ...
0
votes
1answer
15 views

Determining the Likelihoods of Different Game States

Suppose a game is played in which Player 1 must gain two points to win and Player 2 must gain five points to win. Both players start with zero points. In any round, Player 1 has a $1/3$ chance of ...