Conditional Definition and 483 Threads

In probability theory and statistics, given two jointly distributed random variables



X


{\displaystyle X}
and



Y


{\displaystyle Y}
, the conditional probability distribution of Y given X is the probability distribution of



Y


{\displaystyle Y}
when



X


{\displaystyle X}
is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value



x


{\displaystyle x}
of



X


{\displaystyle X}
as a parameter. When both



X


{\displaystyle X}
and



Y


{\displaystyle Y}
are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable.
If the conditional distribution of



Y


{\displaystyle Y}
given



X


{\displaystyle X}
is a continuous distribution, then its probability density function is known as the conditional density function. The properties of a conditional distribution, such as the moments, are often referred to by corresponding names such as the conditional mean and conditional variance.
More generally, one can refer to the conditional distribution of a subset of a set of more than two variables; this conditional distribution is contingent on the values of all the remaining variables, and if more than one variable is included in the subset then this conditional distribution is the conditional joint distribution of the included variables.

View More On Wikipedia.org
  1. T

    Probability of Samples Containing Less than 1.6 mg of Suspended Particles?

    Homework Statement The masses of suspended particles in a sample of water taken from a lake can be assumed to be a random variable which is normally distributed with mean 2.17 and variance 0.979. Find the probability that out of 4 samples of lake water known to contain less than 1.8 mg of...
  2. C

    Proving partitioned matrix involving conditional inverse

    Homework Statement For matrix X partitioned as \underline{X} = [ \underline{X}1 \underline{X}2 ] with \underline{X}1 matrix full column rank rx, Homework Equations prove that \underline{X}(\underline{X}'\underline{X})^{C}\underline{X} =...
  3. N

    Is P a Necessary and Sufficient Cause of Q?

    Hey All, I've just started reading a discrete math book, and in the beginning the book covers logic. One concept I'm finding hard to understand is certain conditional propositions. When the example uses a word problem, I mostly get it. The statement is "If The Mathematics Department...
  4. E

    Calculating Conditional Probability of Y with Exponential RVs

    Dear all, How can I find the the following conditional probability: f_Y(y/\gamma_f\leq \gamma_0), where Y=\gamma_f+\text{min}(\gamma_h,\gamma_g), where all gammas with subscripts f, g, h are i.i.d exponential random variables. Thanks in advance
  5. S

    Relation of conditional and joint probability

    P(A|B)=\frac{P(B|A)P(A)}{P(B)} P(B|A)P(A)= \frac{(P(B)\cap (P(A)) P(A)}{P(A)}=P(B)\cap P(A)=P(A)\cap P(B) P(A|B)=\frac {(P(A)\cap P(B))\vee (P(B)\cap P(A))}{P(B)} 0\leq(P(A)\cap P(B))\leq P(A); P(A)\leq P(B) 0\leq(P(B)\cap P(A))\leq P(B); P(B)\leq P(A)
  6. F

    Conditional Multinomial Problem

    If Y1,Y2,Y2 ~ Multinomial with parameter (n,p1,p2,p3) Prove that the conditional distribution of Y1 given Y3=m (m<n) is a binomial with ( (n-m), p1/(p1+p2) ) p1+p2+p3=1 y1+y2+y3=n y3=m My Attempt: P( Y1=y1| y3=m) = P(Y1=y1, Y3=m)/ P(Y3=m) ( n choose m and y1 ) p1^y1*p3^m / ( n...
  7. N

    Conditional Convergence in a Power Series

    I was wondering if there's an example of a power series \sum_n^\infty c_n (z-a)^n with radius of convergence R so that all z for which |z-a| = R there is purely conditional convergence? (no divergence but also no absolute convergence) Or perhaps a reason why that's impossible?
  8. J

    Symbolic Logic, Proof with Conditional

    Any advice on how to make step 6 check out?
  9. Saladsamurai

    Probability of a Misjudged Innocent Suspect Given Truth Serum Results?

    Homework Statement A truth serum has the property that 90% of the guilty suspects are properly judged while 10% are improperly found innocent. On the other hand innocent suspects are misjudged 1% of the time. If the suspect is selected from a pool of suspects of which only 5% have ever...
  10. K

    Conditional probability and partitions

    Homework Statement I'm currently trying to revise for exams and really struggling on this problem: Suppose you have 3 coins that look identical (ie don't know which is which) with probabilites of 1/4, 1/2 and 3/4 of showing a head. 1. If you pick a coin at random and flip it, what is...
  11. D

    Conditional expectation on multiple variables

    How to compute E[X|Y1,Y2]? Assume all random variables are discrete. I tried E[X|Y1,Y2] = \sum_x{x p(x|y1,y2) but I'm not sure how to compute p(x|y1,y2] = \frac{p(x \cap y1 \cap y2)}{p(y1 \cap y2)} If it is correct, how can I simplify the expression if Y1 and Y2 are iid?
  12. L

    Support of Continuous Conditional Density Functions (Probability)

    f(x,y) = x + y is the joint probability density function for continuous random variables X and Y. The support of this function is {0 < x < 1, 0 < y < 1}, which means it takes positive values over this region and zero elsewhere. g(x) = x + (1/2) is the probability density function of X...
  13. R

    Conditional expectations of bivariate normal distributions

    Hey guys, I'm having a bit of a problem with this question... Homework Statement If X and Y have a bivariate normal distribution with m_X=m_y=0 and \sigma_X=\sigma_Y=1, find: a) E(X|Y=1) and Var(X|Y=1) b) Pr(X+Y>0.5) Homework Equations N/A The Attempt at a Solution...
  14. L

    [PROBABILITY] Conditional probability for random variable

    Homework Statement X and Y two independent random variables with distribution U(0, 1/2). Find the density of (X + Y)2|X - Y > 0 The Attempt at a Solution I was hoping this would be simpler, but somehow I always end up with nothing. The only thing I can work out just fine is that P(X...
  15. D

    Uniform Distribution with Conditional Probability

    Homework Statement So I just took a probability test and I'm having a hard time with the fact that my answer is wrong. I've done some research online and I believe I am correct, I was hoping to get some input. I'm new to using LaTeX so sorry if it's sloppy. Thanks! Problem: Suppose that the...
  16. K

    Conditional & uncoditional MSE (in MMSE estimation)

    Hi, 1- Please explain conditional & unconditional mean square error, and their difference. 2- Which one is the solution for minimum MSE estimation? (that is conditional expectation: E \left[ X|Y \right] . I meant which one is minimized by selecting the conditional expectation.) 3- What is...
  17. T

    Calculating conditional expected value

    Homework Statement Hi All, I have a homework question that I would like some hints on. I am given X and Y with uniform densities on [0,2], and Z := X+Y. The end goal is to find E[X|Z]. Now I know I can use symmetry to argue that E[X|Z] = E[Y|Z], and that E[X|Z] + E[Y|Z] = E[Z|Z] = Z, so...
  18. B

    Basic Probabilities. Conditional Prob.

    Hi, everyone : I have the following problem: We have 3 dishwashers X,Y,Z, with the conditions: 1) X washes 40% of dishes, and breaks 1% of the dishes s/he washes. 2)Y washes 30% of the dishes, breaks 1% 3)Z washes 30% of the dishes and breaks 3%. Question: If...
  19. G

    Sum of independent uniform distribution conditional on uniform

    Homework Statement Let X and Y be independent and normal, then we know that It must be the case that X+Y and X are jointly normal Therefore we can apply the projection theorem: which states that if A and B are jointly normal then VAR(A|B)=VAR(B)-\rho^2VAR(B) , apply the theorem to A=X+Y, B=Y...
  20. B

    Adsolute and conditional convergence of alternating series

    i have a question regarding adsolute and conditional convergence of alternating series. - i know that summation of [ tan(pi/n) ] diverge, but how do we proof it converge conditionally? (ie, (-1)^n tan(pi/n) ] can Leibiniz's theorem be used in this case? but tan(pi/2) is infinite? any...
  21. G

    Understanding Conditional Probability: Formulas and Logic Explained

    Hallow! May anyone please help me on conditional probability i actualy understand how it occurs but the formula? The logic behind it! ie P(A/B)=P(AnB)/P(B)
  22. L

    Conditional probability for random variable

    Homework Statement For the random variable X with the following cumulative distribution function: Calculate P(X\leq1.5|X<2), P(X\leq1.5|X\leq2) and P(X = -2| |X|=2) The Attempt at a Solution This is an exercise about a subject I'm yet to see in class, but the teacher asked us to...
  23. M

    Conditional expectation of exponential distribution.

    I have been stuck at this calculation. There are two exponential distributions X and Y with mean 6 and 3 respectively. We need to find E[y-x|y>x] I keep getting it negative, which is clearly wrong. Anybody wants to try it?
  24. M

    Absolute & conditional convergence

    Homework Statement Determine whether the series converges absolutely, conditionally, or not at all. a) \Sigma (-1)nn4/(x3 + 1) b) \Sigma sin(x)/x2 Homework Equations The Attempt at a Solution a) positive series is n4/n3+1 .. do i do comparison test ?? b) |sin(x)|/x2...
  25. P

    Mathematica Problem with conditional statement in Mathematica

    I am trying to iterate a solution to a non-linear differential equation - but I am unable to evaluate a statement within a Do loop Code: Do[q = Cos[Evaluate[x'[i]/.up]]; If[ q <= 1, Print[q], Print[i], Print[i^2]], {i, 0.1, 1, .1}] x'[i] is a solution to NDSolve The output is i^2, which is...
  26. S

    How Does Independence Affect Conditional Expectations in Probability Theory?

    Question 1) I have X and Y independent stoch. variables What is E[X^2 * Y | X] ? does it generally hold that if X and Y are independent, then every function of X (eg X^2) is independent of Y? Does E[X^2 * Y | X] then become E[X^2|X]*E[Y|X] = E[X^2|X]*E[Y] since X^2 is independent of...
  27. R

    Calculating Conditional Expectation for IID Normal Variables

    If I have x1,x2 iid normal with N(0,1) and I want to find E(x1*x2 | x1 + x2 = x) Can I simply say: x1 = x - x2 and thus E(x1*x2 | x1 + x2 = x) = E[ (x - x2)*x2) = E[ (x * x2) - ((x2)^2) ] <=> x*E[x2] - E[x2^2] = 0 - 1 = -1?
  28. E

    Conditional Probability & Independence

    Homework Statement One satellite is scheduled to be launched form Cape Canaveral in Florida, and another launching is scheduled for Vandenberg AFB in California. Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral...
  29. E

    Conditional Probability of defective components

    Homework Statement Components of a certain type are shipped to a supplier in batches of ten. suppose that 50% of all such batches contain no defective components, 30% contain one defective component, and 20% contain two defective components. Two components from a batch are randomly selected...
  30. R

    Marginal Distribution of X & Conditional Density of λ in STAT134/150

    STAT134/150: Marginal distr. of a random var. w/ random var. param., given distrs Hello, I am a little shaky on my probability, so bear with me if this is a dumb question... Anyway, the distributions of the two random variables are given: X : Poisson (\lambda) \lambda : Exp. (\theta)...
  31. K

    How Does the Network's Structure Affect the Expected Number of Message Forwards?

    Homework Statement An email is sent on the network in which the recipients (0,1,2,3,4,5} are in communication. 1 can send to 4 and 2 2 to 1,3,5 3 to 0,2,5 4 to 1, 5 5 to 0,2,4 0 to 3 and 5 If a message is sent to 2,3,4,5 it is forwarded randomly to a neighbour (even if this means a...
  32. N

    How Can Bit Operations Simplify Conditional Checks in C?

    Homework Statement I want to write the following if statement into simply manipulations using the bit operations: ! ^ & | << >> ~ + Given an x... if ( x == -1 || x == 0) return -1; else return 0; I am dealing with 32-bit integers. How do I go about doing this? (Note: -1 is...
  33. K

    Conditional expectation and partitioning

    Homework Statement I'm told that of n couples, each of whom have at least one child, with couples procreating independently and no limits on family size, births single and independent, and for the ith couple the probability of a boy is p_i and of a girl is q_i with p_i + q_i = 1. 1. Show...
  34. L

    Conditional Dependent Probability

    My questions I am looking at is: We have twelve balls, four of which are white and eight are black. Three blindfolded players, A, B, and C draw a ball in turn, first A, then B, then C. The winner is the one who first draws a white ball. Assuming that each black ball is replaced after being...
  35. R

    [probability theory] simple question about conditional probability

    Hi all, I've got this very simple problem: I know it is an elementary problem, but I never really got into that bayes' theorem, which I need to use here, right? I would be grateful for simple and plain explanation. thanks for your time, rahl.
  36. N

    What is the Conditional Probability of Selecting Box B Given a White Ball?

    box A contains 2 red balls, box B contains 2 white balls, box C contains 1 red ball and 1 white ball, A box is selected at random (with equal probabilities) and one ball is taken at random from that box. (i) Compute the probability of selecting a white ball using Bayes Rule. A(1/3)...
  37. R

    Urn Conditional Probability Problem

    The problem is stated as follows: You have an urn with 12 red balls, 20 green balls, and 13 yellow balls. Suppose 3 balls are drawn without replacement. What is the probability the third ball is yellow given the first ball is green. I am pretty sure I use Bayes formula, but I am not certain...
  38. G

    Conditional expectation of Exp(theta)

    Given X follows an exponential distribution \theta how could i show something like \operatorname{E}(X|X \geq \tau)=\tau+\frac 1 \theta ? i have get the idea of using Memorylessness property here, but how can i combine the probabilty with the expectation? thanks. casper
  39. Y

    Proving X=Y with Conditional Expectation

    How can I do this? Let X,Y r.v., \mathbb{E}(X|Y)=Y and \mathbb{E}(Y|X)=X. Proove that X=Y a.s.
  40. S

    HELP geometric probability: area of a square and conditional probability

    Homework Statement Chose a point at random in a square with sides 0<x<1 and 0<y<1. Let X be the x coordinate and Y be the y coordinate of the point chosen. Find the conditional probability P(y<1/2 / y>x). Homework Equations No clue. The Attempt at a Solution Apparently, according...
  41. H

    Calculating Conditional Probability: A and B Events with A' and B' Notation

    An event A has a 70% chance of occurring, an event B has a 40% of occurring and the event (A n B) has a 20% chance of occurring. The events are not unrelated. If A does not occur what is the probability of B? notation A' means (not A) I have tried to use the formula P(B|A') = P(B n...
  42. N

    What is the probability that the players team wins if he doesn't hit a home run?

    Homework Statement A baseball player compiles the following information : He hits a home run in 34% of his games He gets a strike out in 40% of his games In 78% of his games he hits a home run or his team wins In 10% of his games he hits a home run and gets a strike out In 26% of his...
  43. M

    Finding conditional probability

    Homework Statement There are w white balls and b black balls in a bowl. Randomly select a ball from the bowl and then return it to the bowl along with n additional balls of the same color. Another single ball is randomly selected from the bowl(now containing w+b+n balls) and it is black. Show...
  44. S

    What Is the Probability That Both Children Are Girls Given Specific Conditions?

    Homework Statement A couple has two children. What is the probability that both are girls given that the oldest is a girl? What is the probability that both are girls given that one of them is a girl? DO NOT assume that female births and male births are equally likely. Assume identical...
  45. T

    Conditional Probability, Failure of Concrete Beam

    Homework Statement A concrete beam my fail by shear or flexure. The failure probability in shear is equal to the failure probability in flexure, and the probability of failure in shear when the beam is loaded beyond is flexure capacity (ie, it has already failed in flexure) is 80%. Find the...
  46. R

    Is my Handout wrong? Stumped Conditional Probability

    Conditional Probability... My handout says this... P(m) = .4 P(w) = .5 P(m|w) = .7 Find: P(MnW) = ? P(w|m) = ? P(m or w) = ? P(MnW) = .4 x .5 = .2 P(w|m) = .35/.4 = .875 P(m or w) = ?
  47. B

    Help with conditional probabilty

    I've been struggling for a good couple hours on the below, and I was hoping someone might be able to push me in the right direction... "A" represents the event "the breath analyzer indicates the suspect is drunk" and "B" represents the event "the suspect is drunk." On a given Saturday night...
  48. P

    Why Is the Conditional Entropy of g(X) Given X Zero?

    Hi I'm trying to convince myself that the conditional entropy of a function of random variable X given X is 0. H(g(X)|X)=0 The book I'm reading (Elements of Information Theory) says that since for any particular value of X, g(X) is fixed, thus the statement is true. But I don't...
  49. Repetit

    Conditional convergence of J1(kr)k

    Hey! I need to perform the following integration: \int\limits_0^{\infty} J_1(k r)k dk where J_1(x) is the cylindrical Bessel function of the first kind. This is an oscillatory function with amplitude decreasing as 1/\sqrt{x}. Due to the multiplication of k the integrand however, becomes...
  50. D

    Conditional Probability with combinatorics

    If a hand of 5 cards are dealt from a 52 card pack (order doesn't matter), what is the probability that the hand will contain the ace of spades GIVEN that there is at least one ace? Thanks.
Back
Top