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.
Homework Statement
A particular brand of cars, say ABC, comes in only two colours, white and grey. Exactly
90% of ABC cars in a particular town are white and 10% are grey. Mrs Z, a witness
to a bank robbery, claims to have seen the thieves escaping in an ABC grey car, which
taking into...
Homework Statement
Given f(x) = e^-x and f(y|x) = 1/x e^(-y/x). Three parts: (a) Compute density of (x,y), (b) Compute E(y) and (c) Compute P(y>x).
Homework Equations
f(x,y) = f(y|x)f(x)
if f(x) = ve^(-vx), then E(x)=v^(-1)
The Attempt at a Solution
I'm stuck on a problem. I was...
I'm stuck on a problem. I was given f(x) and f(y|x) and was able to derive f(x,y). The second step of the problem is computing P[y>x]. I think I need to know f(y) to answer this problem but I can't figure out how to derive it. Or is there a way to compute P(y>x) given the info I know without...
Hello everybody,
I have two questions on conditional expectation w.r.t (Polynomial) OLS:
Let X_t be a random variable and F_t the associated filtration, Vect_n{X_t} the vector space spanned by the polynomials of order {i, i<=n }, f(.) one function with enough regularity. I am wondering how...
Hi:
e, z, mu are vectors of size N
I need to show that E(e|z+mu) = E(e|mu) or at least E(e|z+mu) converges in probability to E(e|mu) as N goes to infinity, under the assumption that Z is not correlated with e.
My guess is that to get this result I also need z to be orthogonal to mu...
I'm having trouble with one of the rules of probability
P(A n B) = P(A)P(B) which holds if events A and B are independent
The following problem illustrates my confusion. I've defined Events A and B below, are these events dependent? Per the solution in the book P(A \cap B) = P(A)P(B)...
There are 15 tennis balls in a box, of which 9 have NOT previously been used. Three of the balls are randomly chose, played with, and then RETURNED TO THE BOX. Later, another 3 balls are randomly chosen from the box. Find the probability that none of these balls has ever been used.
Seems...
Homework Statement a family consisting of a father, mother, and a child is chosen at random and is asked on what day of the week each of them was born. What is the probability that all three were born on different days given that the father was born on a monday?
Solution: A is the even all...
I am learning how to prove conditional identities like
(a^2-c^2+b^2+2ab)/(c^2-a^2+b^2+2bc) = (s-c)(s-a)
if a+b+c = 2s
- Derived from Herons formula
I have understood the proof for the above , but i want more problems to work on.
Can anyone suggest some link where i can find similar...
Homework Statement
Suppose a person's score X on a math aptitude test is a number between 0 and 1, and their score Y on a music aptitude test is also between 0 and 1. Suppose further that in the population of all college students in Canada, the scores X and Y are distributed according to the...
This result isn't in our book, but it is in my notes and I want to make sure it's correct. Please verify if you can.
Homework Statement
I have two I.I.D random variables. I want the conditional expectation of Y given Y is less than some other independent random variable Z.
E(Y \...
Please help me with this. Any suggestions are greatly appreciated.
Imagine that I have a bank account. X is the amount of cash on the account at time t+1. Y is the amount of cash at time t. The amount of cash depends on the deposits made and on the amount of cash during the previous period...
This is probably relatively easy, but I'm still a bit confused...
The question:
A family has j children with probability p1 = .1, p2 = .25, p3 = .35, p4 = .3.
A child is randomly chosen. Given this child is the eldest in the family, find the conditional probability that
a) Family has 1...
Homework Statement
boy and girl independently flip each a biased coin with probability of heads p1 for boy, and p2 for girl. they record the number of flips needed until heads shows up.
what is the probability that they tie?
Homework Equations
I just know that we should define a...
Homework Statement
Test the series for (a) absolute convergence, and (b) conditional convergence.
\sum\left(-1\right)^{k+1}\frac{k^{k}}{k!}
Homework Equations
The Attempt at a Solution
So I tried taking the absolute value and then applying the ratio test, which, after...
Hello,
This question relates to Bayes law. I think my problem is I am not sure of the name of the thing I am trying to derive...
I have 2 variables a and b.
a = 1 or 0, b = 0...n
I have the data to calculate;
p(a = 1 and b) p(b)
for any b. Hence I can find p(a=1|b) = p(a = 1 and...
hi all,
I have a question that i really want to know the answer.
the probability that an airplane will return in 10min is 30%, in 20min is 30%, in 30min is 0%,
in 40min is 40% according to the past history records. given that the airplane has not return
in 20min, what is the probability...
I need help about conditional expectation for my research. I get stucked on this point. Could anyone show me how to prove that:
"Let E[|Y|]<∞. By checking that Definition is satisfied, show that if Y is measurable F0, then E[Y|F0]=Y."
Def: Let Y be a random variable defined on an underlying...
I am new in MATLAB and recently, I have tried coding in matlab. So far, I have been getting error msg
"Warning: Failure at t=2.859413e+002. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.015868e-012) at time t."
I have referred...
Hi,
I am a quantum physicist who needs a practical help from mathematicians. :smile:
The physical problem that I have can be reduced to the following mathematical problem:
Assume that we have two correlated variables a and b. Assume that we know all conditional probabilities
P(a|b), P(b|a)...
Dear all,
Please clarify the following:
1.) The difference of conditional probability and Bayes' formula.
2.) Is Bayes' formula a "all weather condition" formula for all conditional probabilities problem?
Thank you,
S
Dear all,
P (A |B) + P (A c|B) = 1 [A c] denotes complement of set A and of course P (B)>0
Is the above statement true?
How about the following two:
P (A |B) + P (A |B c) = 1
P (C ∪ D|B) = P (C |B) + P (D|B) − P (C ∩ D|B)
Homework Statement
In a shipment of 100 televisions, 6 are defective. If a person buys two televisions from that shipment, what is the probability that both are defective?
Homework Equations
the Answer is somewhat weird! it says it is 1/330 ! which is really beyond by recognition...
I ran across this identity for a conditional PDF where the dependent random variable X is continuous and the independent variable N is discrete:
\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}
In the limit as dx approaches 0 this yields:
f_{X|Y}(x|n)...
Hi, I'm trying to work out a probability value from a number of distributions (tests), it gets a little complicated because some of the tests are dependent on each other.
Lets say I have a surface which has black and white regions distributed over it, where black is "true" and white is "false"...
I'm trying to solve a problem as part of my research and it's giving me fits. It seems like it should be simple, but I can't wrap my brain around how to do it. The problem is:
Suppose X~N(0,s), and Y is a random variable that has a probability mass point at 0 but is otherwise uniformally...
Although this problem may look like homework, I assure you it is not. It is a question that arose from a trading card game that I am stuck on. The problem is as follows (with simplified cards)
You have a deck of 53 cards, and 11 of those cards are red and 42 are black. If you were to randomly...
Homework Statement
Let Z_1, \ldots, Z_n be independent standard normal random variables, and let
[tex]S_j = \sum_{i=1}^j Z_i[/itex]
What is the conditional distribution of S_n given that S_k = y, for k = 1, ..., n?
The attempt at a solution
I know that S_j is a normal random...
I'm supposed to calculate the cost of renting a Ford, Cadillac or Toyota, and use conditional statements to calculate the different costs of each vehicle.
users are supposed to enter “F” for Ford, “T” for Toyota, or “C” for Cadillac
users enter “F” for Ford, “T” for Toyota, or “C” for Cadillac...
So we learned about the basic tests for convergence of an infinite series, and we learned about alternating series, and conditional convergence.
Now, I get how to find if a series is conditionally convergent. But what's the use of conditionally convergent infinite series? All we were taught...
The difference btwn marginal distribution and conditional distribution ?
So I have a table that "apparently" shows how a company's employees commute to work.
TRANSPORTATION
JOB CLASS CAR BUS TRAIN TOTAL...
In an election, candidate A receives n votes and candidate B receives m votes, where n>m. Assume that in the count of the votes all possible orderings of the n+m votes are equally likely. Let Pn,m denote the probability that from the first vote on A is always in the lead. Find Pn,m...
[Problem]
Stores A, B, and C have 50, 75, and 100 employees, respectively, 50, 60, and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One employee resigns and this is a woman. What is the prob. she works in store C?
[Solution]
Store A...
Let X, Y be independent exponential random variables with means 1 and 2 respectively.
Let
Z = 1, if X < Y
Z = 0, otherwise
Find E(X|Z) and V(X|Z).
We should first find E(X|Z=z)
E(X|Z=z) = integral (from 0 to inf) of xf(x|z).
However, how do we find f(x|z) ?
[SOLVED] Conditional Expectation
I'm trying to understand the following proof I saw in a book. It says that:
E[Xg(Y)|Y] = g(Y)E[X|Y] where X and Y are discrete random variables and g(Y) is a function of the random variable Y.
Now they give the following proof:
E[Xg(Y)|Y] = \sum_{x}x g(Y)...
conditional probability help please
Homework Statement
Hi there,
I am doing s1 for this jan and i am finding it very difficult to cope up. Especially for probability. I have a cgp buk but stil its not very gud at probability. Here is a question from my text buk which i cud not understand : -...
i need to covert the following conditional statements into logical notation using propositional connectives and quantifiers:
a) A has at most one element
b)A is a singleton
c)ø ∈ A
you don't have to give me the answers, just help me get started or give me some hints
I'm having trouble seeing how this works out. It's blatantly obvious that this is true, but somehow I can't seem to get anywhere on paper with it to simplify it down to anything. Any help would be greatly appreciated!
P\left(A\right|B)=1-P\left(not A\right|B)
Homework Statement
The probability of a monitor not working is 0.005, the probability of a cpu faulty is 0.002, the probability of a keyboard damaged is 0.0025, what is the probability of the computer switching on? If you are then told that the conditional probability of the monitor not...
I am trying to practice for an exam but can't do this question:
does the series \((-1)^n/ln(n) from n = 2 to infinity converge abs/conditionally/diverge?
I know if a do an alternating series test, the integral will converge because lim goes to 0 and a(n+1)<an.
But how can I prove that...
Hi all
First of all, I am new here but I am not new to statistics. But I need your help:smile:
I do have a multivariate normal distribution: x~p(mu,sig)
the vector x has to groups of variables, those that I know are below zero (x_bz), and those that I know are above zero (x_az).
I am...
Homework Statement
Assuming a comp is switched on, the probability that the monitor is not working is 0.005, the probability that the CPU is faulty is 0.02, and the probability that the keyboard cable has been damaged is 0.0025, and that there are no other faults.
Proceed to evaluate the...
I say urgent because of the horribly small lecture I received on this section, a whole 3 minutes or so of examples. While I won't give further context I can say without a doubt I am completely lost.
Here is the problem I am stuck on.
In a string of 12 Christmas tree light bulbs, 3 are...
Homework Statement
Show that \left[\neg\,p\,\wedge\,\left(p\,\vee\,q\right)\right]\,\longrightarrow\,q is a tautology without using truth tables.
Homework Equations
DeMorgan's Laws, etc.
The Attempt at a Solution...
Homework Statement
Let (X_n; for all counting number n) be a sequence of independent random
variables. We focus on the random walk S_n := X_1 + . . . + X_n and set
F_n = 'sigma-algebra' of (S_1, . . . , S_n).
1. Compute E[S_(n+1) \ F_n]
2. For any z belonging to the complex plane C...