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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Heteroskadasticity and Linear Probability

Question Suppose $(Y,X,U)$ be a random vector such that $$ Y = X'\beta + U. $$ Suppose $Y$ takes values in $\{0,1\}$ and that $E[Y\mid X] = X'\beta$. Is it reasonable to assume that $Var[U\mid X]$ ...
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1answer
288 views

Calculate Probability Of Winning At Roulette Using Poisson

Suppose your probability of winning if your number comes up if $\frac{1}{38}$. If so, you win $\$35$, otherwise you lose a dollar. Use the Poisson distribution to show that if you play 70 times then ...
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1answer
579 views

Expectation of throwing $n$ balls into $n$ bins

Suppose we throw $n$ indistinguishable balls in $n$ bins at random. The throws are independent. What is the expected number of empty bins? What is the expected number of bins with one ball. Using ...
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2answers
5k views

Find the P{X < Y} from this joint density function

I have a joint density function given as: $$f(x, y) = 6x^2y$$ $$0\le x\le 1$$ $$0\le y\le 1$$ Now I am asked to find the $P\{X < Y\}$, as well as the $P\{X < 2Y\}$ In order to solve this, ...
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2answers
105 views

How to find the exact value of an upper bound for an exponential random variable

Suppose the waiting time between people entering a store can be modeled by the exponential random variable $X$ with parameter $\lambda=5$. If you use markov's inequality you can find the $P(X\ge 20)$ ...
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1answer
44 views

If $A∩B=A′∩B′$ for $A,A'$ and $B,B'$ independent, do we have $A=A′$ and $B=B′$?

Let $\mathcal{A}$ and $\mathcal{B}$ two independent sigma-algebra, and $A,A' \in \mathcal{A}$, $B,B' \in \mathcal{B}$ such that $A \cap B=A' \cap B' $, and $P(A)P(B)>0$. Do we have $A=A'$ and ...
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2answers
419 views

Does The Monty Hall Problem Still Apply With Infinite Doors?

Here's been a bunch of questions on the Monty Hall problem, so I'll assume people know the basics. This answer helped clarify a few things for me, but talking with some colleagues yesterday, someone ...
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2answers
64 views

Showing that the Geometric distribution $E(X)=\frac 1p$

So I have $X \sim \text{Geom}(p)$ and the probability mass function is: $p(1-p)^{x-1}$ From the definition that: $\sum_{n=1}^\infty ns^{n-1}$ = $\frac {1}{(1-s)^2}$ How would I show that the ...
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1answer
104 views

Probability at a poorly planned gift exchange.

Need help with this problem. I have an idea on a but not sure if a is right nor what to do with b or c. I went to a poorly planned “gift exchange” All five of us tossed our gifts into a bin, and ...
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0answers
191 views

Mellin transform Gaussian

Is there a known formula for the Mellin transform $$M(s) = \int_0^\infty f(x)x^{s-1}\, dx$$ of the Gaussian probability density function $$f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac ...
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1answer
42 views

Simple question regarding probability mass function

X is the number of sixes obtained when a fair dice is thrown 4 times. I just have to write down what the distribution of X is and give the probability mass function. So I have simply written down: ...
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1answer
330 views

What are the chances of winning with a specific card in Spades

In the game of spades, a standard deck is shuffled then all the cards are dealt in a clockwise manner until each of the 4 players has 13 cards. The first play of the game is for each player to throw ...
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0answers
180 views

Improper integral of odd function

I'm a student. In a recent assignment I was asked to find the mean of a Student's t multivariate distribution (which should be $\overline\mu$). I've divided the integral required to find the expected ...
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1answer
77 views

What causes the change in the expected value of the product of random variables?

The following question is part of a homework exercise on portfolio theory that I have to do. Suppose that $Y$ is a random variable representing the returns on an investment. Now, let $f$ be a ...
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1answer
41 views

Does a Markov Blanket allow connections between Parents of a Node?

In a Markov Blanket, we can connect the childredn of a node between them, as a child can be parent (or spouse) of another child. Does this rule apply as well for Parents of a node? In advance, Thank ...
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0answers
86 views

Expectation number of random points exactly on their convex hull

Suppose there are n random points uniformly distributed in a square, what's the expectation of the number of the points located exactly on the edge (or being vertexes) of their convex hull? What if ...
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0answers
109 views

Central Limit Theorem Clarification [duplicate]

The Central Limit Theorem states that the sampling distribution of the sample mean: Converges in distribution to a normal distribution. Has an expected value (mean of the sampling distribution of ...
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1answer
128 views

Hypothesis test question

Can anyone check my answer about hypothesis test? Thanks!
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1answer
177 views

Random variable X has the following discrete distribution

Random variable $X$ has the following discrete distribution: $f(x) = k/x$ for $x = 1, 2, 3$ $f(x) = 0$ otherwise Find $k$ so that $f(x)$ is a legitimate probability mass function What is ...
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2answers
63 views

Can anyone check my answers?

This is about hypothesis testing. Please check my answers Thanks!
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4answers
173 views

Math Expected Value?

We are making a casino game and need to determine expected value to see if we will be profitable. My teacher says our expected value must be a small positive number so that it is fair and that the ...
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1answer
55 views

Seeking hlep on an application of conditional probability

The probability that a store will have exactly k customers on any given day is $P_k(k) = \frac{1}{5}(\frac{4}{5})^K, k = 0, 1, 2, 3, ...$ Everyday, out of all the customers who purchased something ...
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1answer
364 views

Prove that if A and B are independent events then at least one of A and B is either the empty set or the sample space.

Let $S = \{1, 2,..., p\}$, $R$ be the set of all subsets of $S$ and $P(A) = \frac{|A|}{p}$ for all $A \in R$. Suppose p is prime. Show that, if A and B are independent events then at least one of $A$ ...
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1answer
138 views

Variance for mixed distribution (continuous + discrete)

A random variable X has the cumulative distribution function: $F(x)= \left\{ \begin{array}{l} 0 \text{ for x < 1}\\ \cfrac{x^2-2x+2}{2} \text{ for } 1 \le x <2\\ 1 \text{ for } x \ge ...
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2answers
1k views

Can you split up expectation over multiplication?

I was wondering if the property exists where if you have $\ E[(Y- \mu)^3]$ you can write it as $\ E[(Y- \mu)^2] E[(Y- \mu)] $ ?
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1answer
33 views

$95\text%$ confidence interval question

In a study of $205$ adults, the mean heart rate was $75$ beats per minute. Assume the population of heart rates is known to be approximately normal with a standard deviation of $8$ beats per minute. ...
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0answers
46 views

How can I determine this equation?

I have an equation that have been formulated as: $$\Delta w_{ki}=\eta (y_{k}-y_{o})(x_i-x_o)+a_1,$$ where$$y_{k}=\sum_{j}w_{kj}x_{j}+a_{2}$$ and where $a_1,\eta,x_o ,y_o$ and $a_2$ are constants and ...
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1answer
1k views

probability, normal distribution help?

Betsy is testing whether school is more enjoyable when students are making high grades. She asked 150 students if they enjoyed school and whether their GPA was above or below 3.5. She found that 35 of ...
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1answer
90 views

Finding meaningful differences in a set of time measures

An individual is presented with a list of words, one at a time, and asked to answer the first thing to comes to her mind. The delay of all answers is recorded. So I have a list of words, each paired ...
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1answer
38 views

Meaning of estimation?

When we have an estimate of random variable $X$ in terms of random variable $Y$ we can write it as $\hat{X}=f(Y)$ where $\hat X$ is the estimate and $f$ is a function. Then suppose we observe a ...
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1answer
231 views

Probability when a professor distributes a quiz and homework assignment to a class of n students.

Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments ...
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1answer
500 views

Result and proof on the conditional expectation of the product of two random variables

My problem is the following: $X$ and $Y$ are two random variables and $\mathcal{F}$ is a $\sigma$-algrebra. Given that $X$ and $Y$ are independant, and that $X$ is independant of $\mathcal{F}$, can I ...
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1answer
140 views

Limit of Beta distribution on $[0, A]$ as $A\rightarrow \infty$ with constant expectation and variance

I am trying to determine the limiting form of a beta distribution as its range expands under isoparametric constraints on its first two moments.... For reference $X_A \sim Beta(0,A,\alpha,\beta) = ...
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1answer
83 views

A quick chanllenge: height and weight probability problem

The average height and weight of a group of people is 175cm and 70kg; Find the upper bound of the portion of the people who are over 200cm and over 100kg. I thought about Markov inequality, but I ...
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2answers
79 views

A problem on conditional expectation

I am studying conditional expectation and found some problems. I tried to solve them to understand the subject better, but I'm stuck now. Let $X$ be a random variable with strictly positive density ...
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1answer
63 views

Probability Question

The credit department of a clothing shop reported that 30% of their sales are through cash/check, 30% are through credit card and 40% are through debit card. 20% of the cash/check purchases, 90% of ...
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2answers
1k views

Median for continuous distribution

Consider a continuous random variable X with probability density function given by $f(x)=cx$ for $1 \le x \le 5$, zero otherwise. Find the median. First I calculate the CDF: $F(x)=cx^2/2$ for ...
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1answer
84 views

probability problem (infinite repeat)

In this problem $A_1$ ,$A_2$ ,$A_3$ are events and $P(A_1)=t_1$, $P(A_2)=t_2$, $P(A_3)=t_3$, and $S=\{A_1,A_2,A_3\}$. I want to find the probability that event $A_2$ occurs after $A_1$ (or $A_1$ ...
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1answer
76 views

Each player throws two dice

probability problem: players a and b throws two dice and a player wins if the sum for the first throw is 11 or 7 and Each player loses at once if it is 3, 2. For other case,throw two dice is ...
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3answers
132 views

Probability on divisibility

Let S be the set of all 12-digit positive integers each of whose digits is either 1 or 4 or 7 (for example, 477411171747 is a member of S). What is the probability that a randomly picked member of S ...
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1answer
73 views

Finding the median of a probability distribution

A gambler makes a long sequence of bets against a rich friend. The gambler has initial capital C. On each round, a coin is tossed; if the coin comes up tails, he loses 30% of his current capital, but ...
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0answers
75 views

Large deviations rate for binomial distributions

The problem is from Varadhan's Probability Theory, p.39, EXERCISE 3.7. Can you calculate the geometric ratio $$\rho(x)=\lim_{n\rightarrow \infty}\left(\sum_{r\geq nx} ...
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1answer
21 views

Confusion related to the tractability of an integral

I have this confusion related to the tractability of an integral. In the attachment given below for equation 3 why is it intractable. Further in equation 4 they have said that there are $K^n$ ...
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1answer
138 views

Maximum Likelihood

Find maximum likelihood estimator $\hat\theta$ of $f(x;\theta) = (1/2)\exp(-|x-\theta|)$, for $-\infty \leq x < \infty$ and $-\infty \leq x < \infty$. I am confused of how to deal with the ...
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1answer
2k views

Median for Continuous Probability Distribution

Consider a continuous random variable X with probability density function given by: $f(x)=4x(1-x^2)$ for $0 \le x \le 1$ Find the median. So to calculate the median, I calculated the ...
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2answers
78 views

In 1500 trials the correct answer was given 910 times. Is the conjecture plausible?

From Grimmett and Stirzaker's Probability and Random Processes, third edition, section "Two Limit Theorems", problem #2, page 200: It is well known that infants born to mothers who smoke tend to be ...
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1answer
29 views

Probability of a combination

If you have 6 red marbles and 6 blue marbles and 1 white marble, what is the probability of any combination granted it has always has a white marble in a combination of 6 random drawn marble? If you ...
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3answers
104 views

$P\{B^{2}-4AC\geq 0\}$ where $A,B,C \sim U(0,1)$?

The actual problem is to find the probability that $Ax^{2}+Bx+C=0$ has real roots. This boils down to whether or not the discriminant $B^{2}-4AC$ is non-negative. Thus, we seek $P\{B^{2}-4AC\geq 0\}$. ...
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0answers
27 views

Maximal ratio of probabilities

Given events $A, B_1, \dots, B_n$, all with positive probabilities, is there a name for the quantity $$ \max_j \frac{P(B_j\mid A)}{P(B_j)} = \max_j \frac{P(A\mid B_j)}{P(A)} $$ which measures the ...
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1answer
68 views

What is the probability there are two or more claims?

In a group of policy holders for house insurance,the average number of claims per $100$ policies per year is $\lambda=8.0$. The number of claims for an individual policy holder is assumed to follow a ...