Conditional Definition and 483 Threads

I don't get $$\frac{P[x<X<x+dx|N=n]}{dx}=f_{X|N}(x|n)$$ Can someone derive why? I would believe that $$f_{X|N}(x|n)=\frac{f(x,N)}{p_n(N)}$$ but I don't get how that would be the same. And I don't get that $$\frac{P[x<X<x+dx|N=n]}{dx}=\frac{P[N=n|x<X<x+dx]}{P[N=n]}\frac{P[x<X<x+dx]}{dx}$$ Can...

(Mentor note: link removed as not essential to the question.) The problem is: what is relevance anyhow? My questions are these: did I get the math right in the following? Is there a better, more acceptable way to lay out the sample space Ω and the two events F and E? Apart from the math...

Assume a Poisson process with rate ##\lambda##. Let ##T_{1}##,##T_{2}##,##T_{3}##,... be the time until the ##1^{st}, 2^{nd}, 3^{rd}##,...(so on) arrivals following exponential distribution. If I consider the fixed time interval ##[0-T]##, what is the expectation value of the arrival time...

The theorem says The probability that an event B occur after A has already occurred is given by P(B/A) =P(A intersection B) /P(A) But applying thus to a problem like the probability of occurrence of all 3 tails on 3 coins when tossed if 1 tail has already occurred is P(B/A) =(1/8)/(7/8)=1/7...

This question has been driving me crazy. A large industrial firm uses three local motels to provide overnight accommodations for its clients. From past experience it is known that 20% of the clients are assigned rooms at the Ramada Inn, 50% at the Sheraton and 30% at Lakeview. What is the...

Hey! :o Let $P$ be a probability measure on a $\sigma$-Algebra $\mathcal{A}$. I want to prove or disprove the following statements: $P(A\mid B)=1-P(\overline{A}\mid B)$, for $A, B\in \mathcal{A}$ $P(A\mid B)=1-P(A\mid \overline{B})$, for $A, B\in \mathcal{A}$ I have done the following...

In this example: The input multiplier to the artificial neural network, at the bottom of the screen, for each note is determined by an output of an Expectation Maxamazation sequence that converges after ten iterations with input of the artificial neural network's output above a threshhold...

Can someone derive: ##\frac{Cov(Z+\Theta),\Theta)}{\sqrt{Var(Z+\Theta)Var(\Theta)}}=\frac{\sigma ^2}{\sqrt{1+\sigma ^2}}## My attempt: Numerator: ##Cov(X,Y)=E[(X-E(X))(Y-E(Y))]=E[(Z+\Theta-\theta)(\Theta-\mu)]## The denumerator is pretty simple: ##\sqrt{(1+\sigma ^2)\sigma ^2}##

I don't understand what they are doing here. They start with the mgf for the binomial which I understand. But what is ##E[e^{tX}]##? The average of the binomial mgf? And finally why does this explain that X is equally likely to take on any of the values 0,1,..,n?

Dear All sorry for repeated post; There is a problem Problem: Three cards are drawn in succession from a deck without replacement. find the probability distribution for the number of spades. I have come with this solution. Let S1: appearance of spade on first draw S2: appearance of spade on 2nd...

Dear all Please help in solving the following problem. A large industrial firm uses 3 local motels to provide overnight accommodations for its clients. from past experience, it is known that 20% of the clients are assigned rooms at the Ramada Inn, 50% at the sheeraton and 30% at the Lakeview...

Homework Statement Each week, Stéphane needs to prepare 4 exercises for the following week's homework assignment. The number of problems he creates in a week follows a Poisson distribution with mean 6.9. a. What is the probability that Stéphane manages to create enough exercises for the...

Q The amount of time (in minutes) that an executive of a certain firm talks on the telephone is a random variable having the probability density: $$f(x) = \begin{cases} \dfrac{x}{4}&\text{for $0 < x \le 2$}\\ \dfrac{4}{x^3}&\text{for $x > 2$}\\...

Homework Statement Let ##x\in \mathbb{R} ## Prove the conditional statement that, if ## x>-1## then ## x^2 + \frac {1}{x^2+1} \geq 1## 2. The attempt at a solution Suppose ## x>-1## is true. Then ## x^2>1## Then ## \frac{1}{2}>\frac {1}{x^2+1}## Then ##x^2+ \frac{1}{2}>x^2+\frac...

Hi guys, I have a question about computing conditional probabilities of a Poisson distribution. Say we have a Poisson distribution P(X = x) = e^(−λ)(λx)/(x!) where X is some event. My question is how would we compute P(X > x1 | X > x2), or more specifically P(X> x1 ∩ X > x2) with x1 > x2? I...

Homework Statement It rains in a city with a chance of 0.4. The weather forecast is not always accurate. When there will be a rain the next day, the forecast predicts the rain with probability 0.8; When there is no rain, the forecast falsely predicts a rain with probability 0.1. You take your...

Need help with a probability problem. I have the answer from the answer key, I just don't know how to figure it out.An insurance company examines its pool of auto insurance customers and gathers the following information:1) All customers insure at least one car. 2) 70% of the customers insure...

Homework Statement The probability density function for a random vector ##(X,Y)## is ##f(x,y) = 3x##, when ##0 < y< x < 1##. Calculate the conditional probability P(X> \frac{1}{2} | Y > \frac{1}{3}) Homework Equations Conditional probability: \begin{equation} P(A | B) = \frac{P(A \cap...

Homework Statement Manually[/B] trace the following code segments assuming that x equals 12. Show exactly what would be displayed on the terminal. if (x > 0) { printf("x>0\n"); if (x < 10) { printf("x<10\n"); if (x == 12) { printf("x==12\n"); } }...

So I am studying conditionals in proposition logic, and I have discovered that there are a variety of ways to phrase a conditional "if p, then q" in English. Some of the harder ones are... p is sufficient for q a necessary condition for p is q q unless ~p (where ~ is the not operator) p only...

Hey all, I have been doing some math lately where I need to find the conditional expectation of a function of random variables. I also at some point need to find a derivative with respect to the variable that has been conditioned. I am not sure of my work and would appreciate it if you guys can...

Hello, I am confused with the equivalence: (p → r) ∨(q → r) ≡ (p ∧q) → r. I checked that truth tables supports it but I cannot imagine an example which justifies it. Suppose: p says “It is raining”, q says “It is snowing” and r says: “we will close”. So (p → r) ∨(q → r) becomes “if it is...

Hi, I have a function defined by: F[x_,y_,z_]:=3 x+y-2Y[z] where Y[z_]:=3/z I'd like to run Do loop to calculate the value of this function for specific regions of x,y,z. I can use: Do[F[x,y,z],{x,1,2,1},{y,-2,-1,1},{z,3,5,1}], But I have two questions here: First: How to save the output...

Is it possible to solve something like this generally or does it depend on the pdf's of the variables? P(x < f(y) | x > -f(y))

Hey there community, I have a question on an exercise. Actually it is a general question based on it. Here is the exercise: We throw 3 dice. If we know that the sum of these 3 is 10, then what is the probability of at least one of them being 3? Well now, this exercise is very simple. I mean I...

Hi, I don't know how to prove ((Ǝx) F(x) →(Ǝx) (G(x)) with conditional proof from: ((Ǝx) F(x) → (∀z) H(z)) H(a) →G(b) Thanks

Consider three jointly normally distributed random variables X,Y and Z. I know that in the Gaussian case E[Z | X,=x Y=y]=xßZX;Y +yßZY;X where ßZX;Y notes the regression coefficient of Z on X conditional on Y (and ßZY;Xis analogously defined). Is the following derivation correct? E[Z| X>x...

Homework Statement suppose we have 9 balls : 2 red, 3 green, 4 yellow. and we draw 2 balls without replacement, the probability that one of them is red and the other is green is : P(R)P(G\R)+P(G)P(R\G) = (2/9)(3/8)+(3/9)(2/8) i faced a problem in the textbook which says: the probability that a...

Hi, I found this screenshot on a website and I thought it was crazy. I want to calculate the conditional probability of this event occurring because it seems so impossible. Assume NLTH is being played. I want to calculate the conditional probability of this hand being dealt. Here is a...

Homework Statement The problem statement is given below: Homework EquationsThe Attempt at a Solution Here is my attempt so far: I'm sure questions 1 - 4 have been answered. Question 5 is what concerns me. I need to find ##P(C' | D')##, which is the probability a good item is...

Homework Statement Determine ##P(X<Y|x>0)## Homework Equations X and Y are random variables with the joint density function $$ f_{XY}(x,y)= \begin{cases} 4|xy|,-y<x<y,0<y<1\\ 0,elsewhere \end{cases}$$ The marginal densities are given by $$ f_X(x)=2x\\ f_Y(y)=4y^3 $$ The Attempt at a Solution...

Homework Statement If you already have waited five minutes at the bus stop, what is the probability of having to wait additionally five minutes or more? Homework Equations Declare a random time T, together with the CDF shown below, which specify the time (in minutes) that a frozen traveller...

Hello all. I saw this problem in a book. I tried solving it, and compared it to the suggested solution. Results don't match, and I think that I am correct. Could you please help me decide what the right answer is ? This is the question: When coin 1 is flipped, it lands on heads with...

Homework Statement Given X,Y,Z are 3 N(1,1) random variables, (1) Find E[ XY | Y + Z = 1] Homework EquationsThe Attempt at a Solution I'm honestly completely lost in statistics... I didn't quite grasp the intuitive aspect of expectation because my professor lives in the numbers side and...

I'm having trouble understanding an example supposed to motivate Bayes' theorem. Assume that 40% of all interstate highway accidents involve excessive speed on part of at least one of the drivers (event E) and that 30% involve alcohol use by at least one drives (event A). If alcohol is involved...

Homework Statement Suppose that 70% of the statisticians are shy, whereas 30% of the economist are shy. Suppose also that 80% of the people at a large gathering are economists and the remaining 20% are statisticians. If you randomly meet a person at the gathering and the person is shy, what is...

Homework Statement Consider two random variables X and Y with joint PMF given by: PXY(k,L) = 1/(2k+l), for k,l = 1,2,3,... A) Show that X and Y are independent and find the marginal PMFs of X and Y B) Find P(X2 + Y2 ≤ 10) Homework Equations P(A)∩P(B)/P(B) = P(A|B) P(A|B) = P(A) if independent...

So, I know that P ⊃ Q is a true statement even if P is false as long as Q is true. However, I don't understand why that is, or how that is logically sound. Is it because I'm stuck in thinking of these types of statements as "If P, then Q," and they are not supposed to be thought of that way? How...

Homework Statement Suppose you take a pass-fail test repeatedly. Let Sk be the event that you are successful in your kth try, and Fk be the event that you fail the test in your kth try. On your first try, you have a 50% chance of passing the test. P(S1)=1−P(F1)=1/2. Assume that as you take the...

Homework Statement Urn I contains 3 white and 5 red balls, whereas urn II contains 2 white and 1 red ball. A ball is randomly chosen from urn I and put into urn II, and a ball is then randomly selected from urn II. (a) What is the probability that the ball selected from urn II is white? (b)...

Hi all, Let X be a random EDIT variable with (infinite) sample space S. Are there some results dealing with how to maximize E(X|s ) (conditional expectation of X given s ) for s in S ? Thanks.

Homework Statement Selecting Two Balls. Suppose that two balls are to be selected at random, without replacement, from a box containing r red balls and b blue balls. We shall determine the probability p that the first ball will be red and the second ball will be blue I am confusing Pr(A|B) and...

Homework Statement I don't undersand why Pr(A l B) = P(AnB)/P(B) Homework Equations conditional probability[/B] Pr(A l B) = P(AnB)/P(B) A given B The Attempt at a Solution I can make a bunch of problems with this definition intuitively, but I really want to understand with sets what is...

Homework Statement Consider a bag containing 10 balls of which a few are black balls. Probability that bag contains exactly 3 black balls is 0.6 and probability of bag containing exactly 1 black ball is 0.4. Now, balls are drawn from the bag, one at a time, without replacement, till all black...

An electrical system consists of four components. The system works if components A and B work and either of the components C or D works. The reliability (probability of working) of each component is given by P(A) = 0.9, P(B) = 0.9, P(C) = 0.8, and P(D) = 0.8. Find the probability that (a) the...

How many ways to arrange cells of k possible values in a mxn matrix provided that sums of all rows and columns are known? For example, if we have a 5x3 matrix and 10 possible values ( from 0 to 9) that can be assigned for each cell, then how many ways to arrange cells in that matrix satisfying...

Homework Statement We have 6 new and 4 used products in a box. We draw randomly 2 products and used them for a while. then return them back into the box. After that, again we draw 2 products. a) What is probability that at least one is new? b) What is probability that at least one is new if we...

Homework Statement Profesor has 17 keys, and only one unlocks the door. Let random variable X be attempt at which profesor unlocks the door (key that doesn't work is saved by side) Find: a) Most probable value of variable X b) In which attempt will profesor unlock the door c) Probability that...

I know that the following is true and I've already proven it. Let $Y$ be a random variable and $\varphi$ a measurable function. Let $A$ be a $\Sigma_Y$ measurable set. If $ X (\omega) = \varphi(Y (\omega))$ for all $\omega\in A $ , then $\mathbb{E}(X|Y )(\omega) = \varphi(Y (\omega))$ for...

Ok I am really stumped and have no clue what I am missing...Here is the scenario. We have a mother and father who can be either $BB$, $Bb$ or $bb$. The probability for both is:$$P(BB) = P(bb) = \frac{1}{4}$$ $$P(Bb) = \frac{1}{2}$$For the child one gene of the mother and one from the father is...