n*********3
发帖数: 21 | 1 1. Let X-N(0, 1) be a normally distributed random variable with mean 0 and
variance 1. Suppose that x in R, x > 0. Find upper and lower bounds for
the conditional expectation E(X | X >= x)
2. Now suppose that X has a power law distribution, P(X >= x) = alpha*x^(-
beta) , for x >= x_0 > 0, and some alpha > 0, beta > 1. Calculate the
conditional expectation E(X | X >= x), x >= x_0
3. When cycling home at night, I notice that sometimes my rear light is
switched off when I arrive home. Presu | s*******s
发帖数: 1568 | 2 1. [0,inf)
2. beat*x/(beta-1)
3.solve exp(-lamda)cos(i*lamda)=p
(-
【在 n*********3 的大作中提到】
: 1. Let X-N(0, 1) be a normally distributed random variable with mean 0 and
: variance 1. Suppose that x in R, x > 0. Find upper and lower bounds for
: the conditional expectation E(X | X >= x)
: 2. Now suppose that X has a power law distribution, P(X >= x) = alpha*x^(-
: beta) , for x >= x_0 > 0, and some alpha > 0, beta > 1. Calculate the
: conditional expectation E(X | X >= x), x >= x_0
: 3. When cycling home at night, I notice that sometimes my rear light is
: switched off when I arrive home. Presu
| p*****k
发帖数: 318 | 3 few minor additions to solutions by swordmans:
(1) tighter bounds could be got by using eq.(13) on this page:
http://mathworld.wolfram.com/Erfc.html
[ (x+sqrt{x^2+8/pi})/2, (x+sqrt{x^2+4})/2 )
(3) one wants the even n terms, which is hyperbolic cosine,
hence e^(-lambda)[e^(lambda)+e^(-lambda)]/2=p, which gives:
lambda = [log(2*p-1)]/2 | k*******d
发帖数: 1340 | 4 我得出了2的答案，用一般的概率论的方法
但是我有个问题
最近在Stochastic calculus的课上，conditional expectation定义为E[X|G], where
G is a \sigma-algebra. 而conditional expectation 是个random variable,请问如
何将E[X|X>=x]理解成一个随机变量？X>=x 只是一个set吧，{w \in \Omega: X(w
) >=x},它是个严格意义上的conditional expectation吗？我想理解清楚一些
谢谢。 | c**********e
发帖数: 2007 | 5
This solution is great.
This is also great. Only a minor correction: a - sign should be there.
【在 p*****k 的大作中提到】
: few minor additions to solutions by swordmans:
: (1) tighter bounds could be got by using eq.(13) on this page:
: http://mathworld.wolfram.com/Erfc.html
: [ (x+sqrt{x^2+8/pi})/2, (x+sqrt{x^2+4})/2 )
: (3) one wants the even n terms, which is hyperbolic cosine,
: hence e^(-lambda)[e^(lambda)+e^(-lambda)]/2=p, which gives:
: lambda = [log(2*p-1)]/2
| c**********e
发帖数: 2007 | 6
where
Here E[X|X>=x] is not a random variable. If E[X|G] is a random variable,
you must have something random in G. For example, E[X|Y] is a RV since it
depends on rv Y.
【在 k*******d 的大作中提到】
: 我得出了2的答案，用一般的概率论的方法
: 但是我有个问题
: 最近在Stochastic calculus的课上，conditional expectation定义为E[X|G], where
: G is a \sigma-algebra. 而conditional expectation 是个random variable,请问如
: 何将E[X|X>=x]理解成一个随机变量？X>=x 只是一个set吧，{w \in \Omega: X(w
: ) >=x},它是个严格意义上的conditional expectation吗？我想理解清楚一些
: 谢谢。
| m******2
发帖数: 252 | 7 第（3）题能解释一下么？ 谢谢
【在 p*****k 的大作中提到】
: few minor additions to solutions by swordmans:
: (1) tighter bounds could be got by using eq.(13) on this page:
: http://mathworld.wolfram.com/Erfc.html
: [ (x+sqrt{x^2+8/pi})/2, (x+sqrt{x^2+4})/2 )
: (3) one wants the even n terms, which is hyperbolic cosine,
: hence e^(-lambda)[e^(lambda)+e^(-lambda)]/2=p, which gives:
: lambda = [log(2*p-1)]/2
| p*****k
发帖数: 318 | 8
yes, that is correct. thx for pointing it out.
majia222, you want the # of flips to be even to keep the rear light
still on. note sum(n from 0 to infty) lambda^n/n! is exp(lambda),
so in order to get the sum of all the terms with even n, one could
add exp(-lambda) with the same even terms but negative odd terms.
thus the answer is hyperbolic cosine.
【在 c**********e 的大作中提到】
:
: where
: Here E[X|X>=x] is not a random variable. If E[X|G] is a random variable,
: you must have something random in G. For example, E[X|Y] is a RV since it
: depends on rv Y.
|
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