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 ...

learn more… | top users | synonyms (1)

0
votes
0answers
9 views

Mean of a sampling distribution.

Suppose $\hat{p}=1/\overline{X}$ is an estimator of the parameter $p$ of a population variable $X\sim\text{Geo}(p)$. Suppose $p=0.36$ and $n=25$. What is the mean of the sampling distribution? This ...
2
votes
1answer
20 views

Proof of equivalence of two inequalities in Probability

Can you please give me a hint how to prove, that $P(A|B)>P(A)$ and $P(A|B)>P(A|\neg B)$ are equivalent? Thank you I think i have to add, that $0\neq P(B)\neq 1$ and $0\neq P(A)\neq 1$.
-2
votes
0answers
10 views

Consistency and rate of convergence [on hold]

Let $\hat{a}_{n} \geq 0$ be such that $\hat{a}_{n}-a=o_p(1)$ as $n \rightarrow \infty$ with $a >0$. Let $\epsilon_n=o_p(1)$ as $n \rightarrow \infty$. Let $\tilde{a}_{n}:= \max\{\epsilon_n, ...
2
votes
0answers
21 views

Hitting line dartboard?

Assume that a beginner hits a dartboard. What is the probability that you exactly hit the border line between 13 and 6? We were thinking that the probability is zero because the probability of ...
0
votes
0answers
9 views

Stochastic little o, stochastic big o and rate of convergence

I am confused about the relations among stochastic little o, stochastic big o, rate of convergence. In particular: 1) the fact that $a_n-a=o_p(1)$ as $n \rightarrow \infty$ has implications in terms ...
2
votes
3answers
39 views

Probability of $2$ boys in a family.

In a family there are 3 children with minimum $1$ boy.What is the probability there are exactly $2$ boys in the family? I think I have to use combinatorics to solve this problem. I have solved some ...
2
votes
1answer
31 views

Probability conundrum

Good morning, wondered if you could help me please? I would like to work out the probability of and event happening 5 times out of 6. all 6 events have a 1 in 60 chance of a particular outcome. I ...
1
vote
0answers
27 views

Probability Distribution for a Weird Card Game

I promise this is not for a homework problem, even though this sounds like only something a professor would dream up. Here is the game: I begin with a deck of 13 cards: 1 through 10, Jack, Queen, and ...
0
votes
1answer
19 views

Probability - Airplane overselling tickets

Few days ago, I came across a question for probability in one of the interview. Question : The same small commuter plane has 30 seats. The probability that any particular passenger will not ...
2
votes
1answer
13 views

compare $cov(aX, bY)=ab \;cov(X, Y) $to $Var(abX)$ using the marginal distribution $f_X(x)$

I am trying to compare the proof $cov(aX, bY)=ab \;cov(X, Y)$ (which I have already found) to $Var(abX)$ using the marginal distribution $f_X(x)$. I am not sure where to start.
0
votes
1answer
19 views

Distributing work among cupboards

I am self studying, and have the answer to this question at the back of the book. The question is as follows (paraphrase): A survey of chemical research workers has shown on average that each man ...
0
votes
1answer
9 views

Bivariate Continuous Distributions

What is the marginal density of $X$ and $Y$ given the probability density function, ${f(x,y)= \lbrace3x ,\;\;0\le y\le x\le1}$
0
votes
0answers
25 views

Exam grades and bell curve

What is the mathematical explanation for the tendency of exam grades to conform to a bell curve? Initially, I was thinking it should be explained via the central limit theorem, but it's not clear to ...
1
vote
0answers
14 views

Gaussian random vector with 0 mean [duplicate]

Let $X =(X_1,X_2,X_3,X_4)$ be a Gaussian Random Vector with $\mathsf E(X_1)=\mathsf E(X_2)=\mathsf E(X_3)=\mathsf E(X_4)=0$. Show that $$ \mathsf E(X_1 X_2 X_3 X_4) = \mathsf ...
1
vote
1answer
15 views

Poison distribution variance,probability. and mean.

Let $X$ be the poisson random variable such that $P(X = 2) = 9P(X=4) + 90P(X=6)$ a) find the mean and variance of $X$. b) find P(X $\geq 1$) c) find P(X $\leq 10$) Ok so for the first question I ...
2
votes
1answer
9 views

How is this Variance found in this old question?

On this question: Probability: Normal Distribution they find these values: $\hat\mu = .05(150) = 7.5\space,\hat\sigma = \sqrt{150(.05)(.95)} = 2.67$ I see how they got $\mu$, but how did they get ...
-5
votes
0answers
20 views

Gaussian Random vector with zero mean [on hold]

Let $X =(X_1,X_2,X_3,X_4)$ be a Gaussian Random Vector with $\mathsf E(X_1)=\mathsf E(X_2)=\mathsf E(X_3)=\mathsf E(X_4)=0$. Show that $$ \mathsf E(X_1 X_2 X_3 X_4) = \mathsf ...
0
votes
2answers
26 views

ways to roll a die with probability and permutation

how many ways can I roll a die so it is all different numbers every time? ...for instance, what is all the possibilities of rolling all 3 different numbers... so 1 2 3, 2 3 4, etc... but 3 2 1 also ...
1
vote
1answer
16 views

find the mean and variance of this poisson random variable

Let $X$ be the poisson random variable such that $P(X = 2) = 9P(X=4) + 90P(X=6)$ find the mean and variance of $X$. I'm not sure how to approach this problem..am i supposed to multiply each ...
-1
votes
0answers
16 views

Lottery Ticket Probability [on hold]

At a certain retailer, purchases of lottery tickets in the final 10 minutes of sale before a draw follow a Poisson distribution with mean = 15 if the top prize is less than 10,000,000 and follow a ...
1
vote
1answer
19 views

Showing a group of observation is standard normally distributed

Let $X_1,X_2\dots$ be a sequence of independent RVs such that $X_{n}$ is binomial with parameters $2n - 1$ and $1/2$. Define $$Y_{n}=\frac{2(X_{1}+X_{2}+\cdots+X_{n})}{n} -n$$ Show $P[Y_{n}<t]\to ...
0
votes
1answer
14 views

martingales, almost sure convergence

I am given a sequence of independent random variables $(X_n)$ with respective laws given by $P(X_n=-n^2)=\dfrac{1}{n^2}$ and $P(X_n=\dfrac{n^2}{n^2-1})=1-\dfrac{1}{n^2}$, and letting $S_n=X_1+...+X_n$ ...
0
votes
2answers
29 views

What is $E[||X|-2|]$? The expected value of an absolute value

What is $E[||X|-2|]$? Can I just get rid of the $2$ and continue or should I look at two different values, one for greater than $0$ and the other less than $0$?
-2
votes
0answers
10 views

Random Sample taken [on hold]

A random sample of 300 people are taken. What is the probability that at least 100 of them are over 180cm in height given average height = 175 and standard deviation = 10?
-1
votes
0answers
13 views

Normal Distributions

The average height for females is 175cm with standard deviation 10. a) What is the minimum height a woman can be to be in the top 5% of this population? I solved this by finding the z-score that ...
1
vote
2answers
29 views

Can 2 different random variables have the same CDF?

I'm looking for proof that two different random variables can have the same Cumulative Distribution Function; in other words, I'd like to disprove that a CDF uniquely defines a random variable. ...
1
vote
1answer
34 views

Can you solve this Halloween raffle probability problem?

Five raffle tickets are sold for a Halloween raffle. Marty buys three of the five raffle tickets. Two of the tickets are to be drawn as winning tickets. What is the probability of Marty not having a ...
0
votes
2answers
23 views

Co-relation Coefficient

$X$ and $Y$ are jointly continuous random variables. Their probability density function is: $$f(x,y) = \begin{cases}2x & \mbox{if } x\in [0,1], y\in[0,1] \\ 0 & \mbox{ otherwise ...
0
votes
1answer
16 views

Finding the probability of a randomly selected event?

I know I'm over-thinking the following question, I just need to know how to start! In a certain population of women 4% develop symptoms of a classic disease, 20% are smokers, and 3% are smokers and ...
2
votes
1answer
20 views

Exponential distribution - maximum earthquake magnitude

Suppose $n$ earthquakes occur, and suppose the magnitude of earthquakes are independent and have an exponential distribution with mean $1$. What is the pdf of the maximum earthquake magnitude?
1
vote
1answer
19 views

A Question about Shuffling a Deck of Cards

Currently I am following Sheldon Ross' A first course in probability. And I got stuck in this question: Consider the following technique for shuffling a deck of $n$ cards: For any initial ordering of ...
11
votes
4answers
1k views

Can you pick a random natural number? And a random real number?

Is it possible to pick a random natural number? How about a random real number? Is the axiom of choice involved in this?
1
vote
1answer
24 views

Joint probability density function probability

$X$ and $Y$ are jointly continuous random variables. $$f(x,y)=\begin{cases}kx & x\in[0,1], y\in [0,1]\\0 & :\text{otherwise}\end{cases}$$ a) What value of $k$ makes this a density ...
0
votes
1answer
19 views

Represent probability with multiple distributions. Archer shooting bullseyes problem.

The goal is to come up with two ways to represent this probability: An archer shoots a bulls-eye with probability $0.4$. If the archer shoots ten arrows, what's the probability that at least 3 are ...
2
votes
2answers
39 views

probability question with proof type question

If $P (A \cap B) > P (A)P(B)$, then events A and B are positively correlated. Show if $P(A\mid B) > P(A\mid B^c)$, then $A$ and $B$ are positively correlated. I know $$P(A\mid B^c) = ...
2
votes
1answer
21 views

Expectation of vector valued functions

Let $t_1,\ldots,t_m$ be $m$ random variables that are independently and identically drawn from a Bernoulli distribution with a constant parameter $p$. Now, we define some functions of ...
0
votes
1answer
15 views

Limit of a jointly independent sequence of random variables

Can I say the following? If a jointly independent sequence of random variables $X_1,X_2,\dots$ converges to random variable $X$ in the mean square sense, then $X$ is independent of the elements of ...
0
votes
1answer
22 views

How to Find joint Probability mass function of $P_{AB}(a,b)$ given $P_A(a)$ and $P_B(b)$?

A Super market orders supplies by fax. Depending on the size of the order, a fax can be either: Page for a short order pages for a long order The market has three different suppliers: The cheese ...
2
votes
1answer
31 views

Find the value of k which makes f a density function.

Observe the following probability density function for a continuous random variable X $$f (x) = \begin{cases} k\sqrt x (1-x) &\text{ for }x\in(0,1)\\ 0 &\text{ otherwise} \end{cases} $$ Find ...
0
votes
0answers
50 views

Summation of product of two binomial probabilities

I am trying to find the closed form solution for this formula but got stuck: $\displaystyle\sum_{k=m}^{\infty}{\binom{k}{m}\cdot2^{-k}}$ Actually I try to compute the values of summation of product ...
0
votes
1answer
22 views

constructive counting math problem about checkers on a checkerboard

In how many ways can we place anywhere from 0 to 9 indistinguishable checkers on a 3x3 checkerboard (no more than one checker per square), such that no row or column contains exactly 1 checker? I ...
2
votes
2answers
19 views

Expected value of X-x for exponential distribution

Assume $X\sim$ exponential$(\lambda)$. In class we noted that $E[X-x|X\geq x]=\frac{1}{\lambda}$. Why is this? I would have thought that $E[X]-E[x]=\frac{1}{\lambda}-x$.
0
votes
1answer
30 views

Drawing Probability Density Function

Can someone help me to draw this pdf? I really don't have an idea how to convert a function to pdf. Thank you p(x | c) = 1/3 for 1 <= x <= 4 and P(c) = 0.5
3
votes
1answer
48 views

Normal Distribution and Cofffee

For my homework I have this question: A coffee vending machine automatically pours different types of coffee into cups. The amount of coffee dispensed is modeled by a normal distribution with mean ...
1
vote
0answers
19 views

r successes contained in a set of s trials in a sequence of n Bernoulli trials

Considering a sequence of n Bernoulli trials, I'm looking for the probability to obtain at least r consecutive successes within a set of s trials. Meaning, there must be at most s - 2 trials between ...
1
vote
0answers
27 views

Probabilty Models and distribution techniques

Coliform bacteria are distributed randomly and uniformly throughout river water at the average concentration of one per twenty cubic centimeters of water. Part (c) In testing for the concentration ...
0
votes
2answers
16 views

Probability of 7 straight roulette wheel spins within same group of 12 numbers

What are the odds of 7 straight spins on a roulette wheel rendering a number within the same group of 12 numbers? (i.e. 7 numbers within 1-12, 7 within 13-24 or 7 within 25-36). This would be a ...
3
votes
1answer
55 views

Selecting a Random Point Inside a Cube

A point $P$ is selected at random inside a cube. Find the probability that $\angle APB \geq 135^o$, where $\overline{AB}$ is a body diagonal of the cube. I am not able to come up with the right ...
1
vote
2answers
24 views

For Continuous RVs $X$ and $Y$ if $Y=$ “the CDF of $X$ evaluated in $X$” why does that mean $Y$ is uniform over (0,1)?

Let $X$ be a continuous random variable with CDF $F$. Define the random variable $Y=F(X)$, show that $Y$ is uniformly distributed over $(0,1)$. I have literally no idea why this would even be ...
0
votes
0answers
27 views

Carnival Shooter Puzzle

Bet: Being at the carnival shooter, Randy proposes a bet to Susan. Whoever shoots $x=80g$ of plastic stars first wins. The loser pays. Setup: The shooting board features $k$ plastic stars with a ...