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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0answers
4 views

What is the probability in the following case?

Given 100 cells, each cell can contain the values 0 or 1 with 0 - with a probability of 0.96 1 - with a probability of 0.04 How can I calculate the probability ...
0
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0answers
7 views

Poisson Distribution word problems

During rush hour the number of cars passing through a particular intersection23 has a Poisson distribution with an average of 540 per hour. (a) Find the probability there are 11 cars in a 30 second ...
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0answers
9 views

Introducing a sequence of constants approaching zero in the Slutsky's Theorem

Let $X_i$ be a random variable. Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $ \mathbb{E}(X_i)=\mu$ and $Var(X_i)=\sigma^2>0$. Let ...
0
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0answers
12 views

Probability of accessibility

Between A, B, and C, there are the following highways: A – B, A – C, and B – C. During monsoon, when there is heavy rain, each of the road gets blocked independently with probability $p$. What is ...
0
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0answers
9 views

Find the distribution function for random vector and what if it has discontinuities

Let $X,Y$ be random variable. Suppose $Z:=(X,Y)$ be random vector and $P(X=Y)=1$ and $P(a \le X \le Y) = b-a$ with $0 \le a \le b \le 1$ Question 1: what is the distribution function of $Z$? ...
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3answers
31 views

Find the pdf of T = X + Y

Let (X,Y) be a random point chosen uniformly on region R = {(x,y) : |x| + |y| <= 1}. I need to find the pdf of T = X + Y. I know the joint density is just equal to 1/(area) = fxy(x,y) = 1/2 for ...
0
votes
1answer
26 views

Probability: What ist the expected value of the number of comparisons made during a linear search?

I would like to propose the following problem: A linear search is performed to check whether a given name 'N' is on a list. The list contains 10 distinct names. The details: The search begins by ...
0
votes
1answer
12 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
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1answer
24 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
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0answers
25 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 ...
2
votes
3answers
44 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
35 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
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0answers
28 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
23 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
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1answer
20 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
10 views

Bivariate Continuous Distributions

What is the marginal density of $X$ and $Y$ given the probability density function, ${f(x,y)= \lbrace3x ,\;\;0\le y\le x\le1}$
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0answers
29 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
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0answers
18 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
16 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 ...
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0answers
26 views

Gaussian Random vector with zero mean

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
17 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
15 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
30 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
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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
18 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
22 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 ...
12
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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
26 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
41 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
16 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
32 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
53 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
23 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
20 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
31 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
2answers
60 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
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0answers
21 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 ...