All Questions
Tagged with expected-value conditional-probability
73
questions
6
votes
2
answers
176
views
Inequalities involving expectations
Consider four random variables $W, X,Q,Y$, where $Q$ and $Y$ are binary. Assume
$$
\begin{aligned}
& (1) \quad E(Q(X+W)|Y=1)\geq 0\\
& (2) \quad E(W|Y=1)=0\\
& (3) \quad \Pr(Q=1|Y=1,W)=1
\...
- 889
0
votes
0
answers
27
views
Why $\mathbb{E}_{(x, y) \sim \mathcal{D}}[f] = \mathbb{E}_{x \sim \mathcal{D}_{X}}[\mathbb{E}_{y \sim \mathcal{D}_{Y|x}}[f|X=x]]$ [duplicate]
I found this equality on p.6 in this document proving that Bayes Predictor is optimal (i.e. it achieves the minimal generalization risk) amongst al hypotheses:
$$
\mathbb{E}_{(x, y) \sim \mathcal{D}}[\...
- 720
2
votes
0
answers
96
views
What is the expected length of an interval on an arc of a circle that can be constructed using exponential variates?
I had asked this question on Math stackexchange once before and now again but this does not seem to be drawing too much attention. Since this is a question that can be safely classified as non-measure ...
- 71
0
votes
0
answers
137
views
Is a nested expectation equivalent to expectation on the joint? $\mathbb{E}_{p(x)}[\mathbb{E}_{p(y \mid x)}[f(x, y)]] = \mathbb{E}_{p(x, y)}[f(x, y)]$
Is it always true that nested expectations equal expectations on the joint?
$$
\mathbb{E}_{p(x)}[\mathbb{E}_{p(y \mid x)}[f(x, y)]] = \mathbb{E}_{p(x, y)}[f(x, y)]
$$
Something along these lines ...
- 657
0
votes
0
answers
51
views
Derive E[Y|X] when the joint probability is given
Now, consider joint density of $X, Y$ :
$$
f_{X, Y}(x, y)=\left\{\begin{array}{l}
\frac{1}{\pi} ; X^2+Y^2<1 \\
0 ; \text { Otherwise }
\end{array}\right.
$$
Derive $E(Y \mid X)$.
I know how to ...
- 25
2
votes
2
answers
189
views
Is the expected value of a probability over an interval meaningful?
I am reading an unpublished manuscript and have come across an equation of the following form for the calculation of the probability of an even A,
$$
P[A]=E\Big[P[X>x|Y]\Big]. \tag{1} \label{1}
$$
...
- 123
0
votes
1
answer
158
views
Conditional survival function in landmark analysis
In H.Putter & H.C. van Houwelingen's paper
"Understanding Landmarking and Its Relation with Time-Dependent Cox Regression"
the authors state that the conditional survival function, given ...
- 5
1
vote
1
answer
164
views
Expected number after n rounds of uniform~[0,1] draws
If we have a series of $n$ IID random variable $X_i$ that are uniform [0,1], and at each round $i$ we decide to either keep $X_i$ or discard it for the next number. What is our strategy to maximize ...
- 321
1
vote
0
answers
150
views
"Linearity" of the Normal Distribution
I am trying to understand the following statement:
Can someone please explain what is meant by "the conditional expectation function m(x) is linear in x"?
In the case of regression, I ...
1
vote
0
answers
161
views
classical m balls and n bins problem but more tricky
This is a problem related with the classical $n$ bins and $m$ balls problem but has some modifications that makes it more tricky to solve:
In this case, the probabilities of the bin´s system are not ...
- 11
10
votes
2
answers
193
views
A farmer is growing a magical tree
This is not homework. It's a story I came up with to explain a statistical distribution I became interested in. If this is a known distribution, I'd love to be pointed in that direction.
A farmer has ...
- 141
1
vote
0
answers
57
views
Computing the expectation of a product from the conditional expectation
Consider 3 random variables $Y, W,X$. Suppose that we know
$$
(1) \quad \mathbb{E}(Y| X=x, W=w) \quad \text{ for each possible values $x,w$ taken by $X,W$}
$$
Question: Can we compute from such ...
- 889
1
vote
0
answers
20
views
2 boxes of money, unknown money inside, one is twice the amount of the other [duplicate]
Suppose, I have two boxes. A host is telling me to pick one. I see the money inside of this one - say 100 \$. Now, I need to decide if I want to swap, given the information that the other box has ...
- 11
0
votes
1
answer
361
views
Conditional expectation versus correlation
Consider two random variables $X$ and $Z$. Suppose $E(X)=3$ and $E(X|Z=z)=0$ for some realisation $z$ of $Z$.
Does this imply that $X$ and $Z$ are correlated?
Does this imply that $X$ and $Z$ cannot ...
- 889
2
votes
1
answer
144
views
Derivation of expected loss ESL (integrating over conditional expectation confusion)
I am trying to understand the derivation of expected loss (equation 2.11 in Elements of Statistical learning) and there is a specific step I do not understand.
We start with
$EPE(f) = E(Y - f(x))^{2}$
...
- 341