Conditional Definition and 483 Threads

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Hi, I am teaching a machine learning course and the students have very poor knowledge about conditional probability, Bayes rule etc. Most students have done their undergraduates years ago and I guess their educational background has not been that good. Last lecture was on Naive Bayes...

Suppose X and Y are independent Poisson random variables with respective parameters λ and 2λ. Find E[Y − X|X + Y = 10]3: I had my Applied Probability Midterm today and this question was on it. The class is only 14 people and no one I talked to did it correctly. The prof sent out an e-mail saying...

Homework Statement A teacher would like to distribute 20 candies to 5 children, each of which receives at least two candies. (a) Find the probability that at least one child receives at least 6 candies. (b) Find the probability that at least one child receives at least 7 candies if at least...

Homework Statement A company has been running a television advertisement for one of its new products. A survey was conducted. Based on its results, it was concluded that an individual buys the product with probability...

Homework Statement Santa has n types of presents. Every child can receive at most one present of each type and: a) every child has to get a present AND cannot receive the same set of presents as any other child. b) for every 2 children, there must be a present that both of the children get How...

I know when a series is conditionally convergent and I understand that being conditionally convergent means that rearrangement of the terms will not always lead to the same sum, but I am unsure why exactly this is important? I have not really experienced a time where I am rearranging terms...

Hello again. I recently submitted a thread asking for feedback on a couple of very basic proofs for an exercise from the book "How To Prove It" by Velleman. This is another request for you to help me understand how wrong my proof for a new exercise could be improved. 1. Homework Statement...

So I'm trying to find E(X1,X2|Xn) where X1,X2,...Xn are the numbers of cell observations in a multinomial distribution. How do I even calculate this? I know it is not independent so I cannot split it. Does it have something to do with the fact that E(Xi)=nPi ?

Hi, I'm trying to show that Givien a probability triplet (\theta,F,P) with G\in F a sub sigma algebra E(E(X|G))=E(X) Now I want to use E(I_hE(X|G))=E(I_hX) for every h\in G since that's pretty much all I've for the definition of conditional expected value. I know this property should use the...

##P(A|A∩B) = \frac{P(A∩(A∩B))}{P(A∩B)} = \frac{P(A∩B)}{P(A∩B)} = 1## So given the the event "A and B" as the sample space, the probability of A occurring is 1. ##P(A|A∪B) = \frac{P(A∩(A∪B))}{P(A∪B)} = \frac{P(A)}{P(A∪B)}## Those two events are independent if and only if the probability of "A or...

Hi, I've been wondering recently what types of systems/processes can give rise to independence. E.g. 'a' and 'b' are independent only given that the system constraints exist. I'm coming from biology so by independence I don't necessarily mean the strict mathematical version but something like...

Homework Statement Consider the series: \sum\limits_{k=17}^\infty (-1)^{k}(\sqrt{k-3}-\sqrt{k-5}) Homework EquationsThe Attempt at a Solution First, I will attempt to determine whether it is absolutely convergent: \lim\limits_{k\to\infty} \left(\sqrt{k-3}-\sqrt{k-5}\right) = 0 Since the limit...

Hey guys, I'm trying to find a conditional distribution based on the following information: ##Y|u Poisson(u \lambda)##, where ##u~Gamma( \phi)## and ##Y~NegBinomial(\frac{\lambda \phi}{1+ \lambda \phi}, \phi^{-1})## I want to find the conditional distribution ##u|Y## Here's what I've got so...

Hey guys, I'd love to get the formula to answer the following question: I have 50 colorless balls and 100 different paints. All balls have to be painted. Note that the balls' order is not important and repetition is allowed (means I can paint up to 50 balls in any single color if I want). With...

P(A/B) is defined to be P(A∩B)/P(B) Why is this true? When A and B are dependent events, I can understand why this is correct. It is clear when you see the venn diagram. But for independent events, why is the formula correct? Any intuition or formal proof?

am a bit confused, if i want to find out for example the P(Having the disease among everyone) , using conditional, would it be total people Having the disease over total population?Prescreening Positive and Have the disease is 66 Prescreening Positive but does not the disease 150 prescreening...

Basically I am wondering how you deal with a conditional cdf and turning that into a conditional pdf when the random variables are independent. I know that f(X|Y) =f(X)f(Y)/f(Y)=f(X) I tried to derive this in a nice attached laTex document but it does not seem right to me. Note(this is for a...

Homework Statement Let X and Y be independent Bernoulli RV's with parameter p. Find, \mathbb{E}[X\vert 1_{\{X+Y=0\}}] and \mathbb{E}[Y\vert 1_{\{X+Y=0\}}] Homework EquationsThe Attempt at a Solution I'm trying to show that, \mathbb{E}[X+Y\vert 1_{\{X+Y=0\}}] = 0 by, \begin{align*}...

So say I have Pr(Z|X&Y) I'm guessing that it follows the standard Pr(A|B)=[Pr(B|A)Pr(A)]/Pr(B) So Pr(Z|X&Y)=[Pr(X&Y|Z)Pr(Z)]/Pr(X&Y)?Also, if X&Y are independent, then would I get Pr(X&Y|Z)=Pr(X|Z)Pr(Y|Z)?

Homework Statement If I have a given in a proof in the form: A or B or C ... etc. etc. and if I choose to approach this given in a case by case basis: (assuming one of the A,B,C... one at a time) and if one or more of the assumptions contradicts some other given in the proof does that mean that...

Homework Statement Let X and Y be independent exponential random variables with parameters a and b. Calculate E(X|X+Y). Homework EquationsThe Attempt at a Solution I'm pretty sure I have it, just want to make sure. Joint density for X and Y is abe^(-ax)e^(-by) for x,y>0. Let Z=X and W=X+Y so...

It is known that if it rains, there is a \(50\%\) chance that a sewer will overflow. Also, if the sewer overflows, then there is a \(30\%\) chance that the road will flood. If there is a \(20\%\) chance that it will rain, what is the probability that the road will flood? Let A be the...

Homework Statement Suppose there are two urns that contain white and yellow balls. Urn 1 contains 10 white and 5 yellow balls, and Urn 2 contains 6 white and 12 yellow balls. You are going to draw 3 balls without replacement from one of the urns. To decide which urn to draw from, you will...

1. I have a problem that I cannot figure out how to solve. I want to find the following: E(X|X<Y) where X follows exp(a) and Y follows exp(b) (exp is for exponential distribution). Any ideas on how to solve it? [b]I got E(X|X<Y) = \int_{-∞}^{∞} E(X|X<y)f_{y}(y)dy = \int_{-∞}^{∞}...

Hello all, I've been stuck on this question for a while and it's annoying the stew out of me! I know it's a basic definition type of question, but I can't seem to understand it. Can any of you help? Question: Let X be a random variable and A be an event such that, conditional on A, X is...

I have a group of dogs (3 brown male, 2 brown female, 4 white male, 4 white female, 5 black male, 4 black female) What is the probability to 1. select a female brown dog ? 2. select a female, given that is a brown dog ? 3. select a brown given that is a female dog ? Thank you. I have...

Homework Statement We've got a standard deck of 52 cards. We shuffle the deck well, then cut it into two piles of 26 each, a top pile and a bottom pile. We reach in and pull a random card out of the top pile, observing that it is an Ace. We then put it into the bottom pile, shuffle the...

Dear All, I am a starter to machine learning and i am currently confused about the following problem: what is the result of P(X|Y)P(Y|Z)? In my book, it is written to be P(X|Z). But I don't think it is correct since P(X|Z)= P(X|Y,Z)P(Y|Z) But clearly P(X|Y)=/= P(X|Y,Z) Assuming...

I'd like some help understanding a proof, from http://www.statlect.com/cndprb1.htm. Properties are introduced, which a conditional probability ought to have: 1) Must satisfy properties of probability measures: a) for any event E, 0≤P(E)≤1; b) P(Ω)=1; c) Sigma-additivity: Let {E1, E2, ...

I really need some help here, will appreciate any effort. I calculated time series of tidal stresses. It turned out that the probability of having positive tidal stress is 0.4 and negative - 0.6 (I counted up number of hours when the stress was positive/negative and divided by the total number...

Homework Statement Often times, a program accepts input from a user, and needs to check the validity of the input, and then produce useful and informative error messages if the input is invalid. Suppose that uiVal is a variable representing the user's input, and that errorCode is an 1-by-0...

Homework Statement #1 in the attachment Homework Equations The Attempt at a Solution My code is working for all of the numeric, logical, character portions. I got 7/8 points if ischar(X_input) Y_output = upper(X_input); elseif isnumeric(X_input) switch...

Homework Statement Using the switch-case construction, write code that take a variable named Shape containing a string and assigns to the variable numSides the number of sides of the shape named in the variable Shape. Your code should be able to return the number of sides for a triangle...

I know that, for a random vector (X,Y,Z) jointly normally distributed, the conditional expectation E[X|Y=y,Z=z] is an additive function of y and z For what other distributions is this true?

I'm reading a journal article that implies the following but I can't see how it is done. I'll give 100 DogeCoin (or equivalent) to whomever can explain this in full. Given that V(A|B) = s V(A) = r*s + w B = A + C and A & C are independent so V(B) = V(A) + V(C) & V(C) = V(B) - V(A) Then how...

let X1 and X2 be independent Poisson variables with respective parameters μ1 and μ2. Let S = X1 + X2. Is X1 given S=s a binomial dsitribution? What is the parameters? I just can show that S is a Poisson with mean μ1 + μ2. But I am not confirm X1 given S is a binomial or not? Someone please...

Consider a family of densitites $f(x,\theta)=\frac{exp(-{\sqrt{x}})}{{\theta}}$. Let $X_{1}$ be a single observation from this family. I have shown that ${\sqrt{X_{1}}}/2$ is an unbiased estimator. Now consider $n$ observations $X_{1},..X_{n}$. I have shown that...

Hello. I was wondering if the following is correct. Let S= a*X/(b*X+c), where a,b,c are positive constants and X is a positive random variable. Also let H= h, where h is also a positive random variable (S and H are mutually independent). Then, let F_{Z}(.) and f_{Z}(.) denote the CDF...

Hi guys, I am having trouble showing that \mathbb{E}\left[(Y-\mathbb{E}[Y|X])^{2}\right]=0. I understand the proof of why E[Y|X] minimizes the mean square error, but I cannot understand why it is then equal to zero. I tried multiplying out the square to get...

This is a fun question i found on the internet, a bit harder than my course and I've spent hours on it but can't find a solution, i was hoping someone could help me. Here's the situation. You are in jail and have been sentenced to death tomorrow, however there's a way out. you're given 12 red...

Homework Statement x,y,z are three terms in GP and a,b,c are three terms in AP prove that (xb÷xc)(yc÷ya)(za÷zb)=1Homework Equations The Attempt at a Solution (xb-c)(yc-a)(za-b) since x y z are in GP xb-c÷yc-a=yc-a÷za-b (xb- c)(za-b)=yc-a(yc-a)

1. The pratement, all variables and given/known data If a =1÷(1-b) ,b=1÷(1-c),c=1÷(1-d) prove that a=d Homework Equations The Attempt at a Solution a=1÷(1-b) a-1÷(1-b)=0 {a(1-b)-1)}÷1-b=0 a-ab-1=0 a-ab=1 similarly b-bc=1 c-cd=1 could any of you please give a hint, this was...

Homework Statement If random variables X and Y are independent and both belong to Possion distribution of parameters \lambda_1 and \lambda_2 , then what is the conditional distribution of X when the condition X + Y = m is given? Homework Equations Possion distribution of...

A conditional probability problem that I can't figure out: (PART 1) Tom is trying to figure out the probability that his memory of a ride in a fighter jet when he was a toddler is real or if it is just a product of his imagination. He grew up near an air force base -- the odds of a child in that...

Homework Statement The problem is as follows. I have two spins, m_S and m_I. The first spin can either be \uparrow or \downarrow , and the second spin can be -1, 0 or 1. Now, I envision the situation as the first spin being on the bloch sphere, with up up to and down at the bottom. What I...

Hi all, I am reading Suppes' book on axiomatic set theory and having difficulties understanding the part on formal conditional definition. Background in p.18, he gave the rule for operator conditional definition as follows: An implication P introducing a new operation symbol O is a...

Homework Statement Find the conditional distribution function and density for the random variable X defined on R given that X is in some interval I = (a,b) where P(X in I) > 0. Assume that the density and distribution for the random variable X is known Homework Equations fX|X\inI =...

Homework Statement There are two types of cooking oil, mono- and polyunsaturated. In a supermarket, 10.526% of the oil sold is mono-, of this 3.684% is canola oil and 6.842% is corn oil. The remaining 89.48% of the oil sold is poly-, of this 48.95% is canola oil and 40.53% is corn oil...

Homework Statement A new diagnostic test is developed in order to detect a particular disease. It is known that 1% of the population has this disease. The diagnostic test is said to be 95% reliable. In other words, if a person has this disease, the test will detect it 95% of the time. On...