no integration needed!Pr(X_(1) > x) = Pr(x_1 > x) * ... * Pr(x_n > x) = exp(-(\lambda_1 + ... \lambda_n) x)X_(1) is exponentially distributed with rate parameter of the sum of ....

<t>I think SPMars is right, but the answer is not unique, ie, this will work as well:(2,3,3,4,4,5) and (0,2,3,4,5,7)The trick is to realize that for a pair of fair dice, the probability of the sum being k is the coefficient of x^k of the following expansions:(x+x^2+x^3+x^4+x^5+x^6)^2/36You can re-ar...

How can a volatility be normally distributed? (\sigma > 0)

<t>Here is one way to solve this:Let L(N) be the length of sequence when you draw N iid r.v., thenL(N) = L(N-1) + I_{X_n = X_(n)} (X_(n) is the n-th order statistics)Take expectation:E[L(N)] = E[L(N-1)] + Prob (X_n = X_(n)) = E[L(N-1)] + 1/n (since X_1, X_2, ..., X_n are iid)E[L(1)] = 1so E[L(N)] = ...

<t>If randomly jumping on the train seems confusing, we can re-frase the question as follows:There are n(n is unknown) balls in a bag, the balls are numbered 1 through n, now randomly pick a ball from the bag, you see it's numbered 10, what is the expected number of balls in the bag?This is similar ...

<t>Let B(t) be a Standard Brownian Motion, with B(0) = 0 a.s.Is there a closed form solution for Prob(B(t) hits a before hits b), for any real numbers a and b?I know you can use reflection principle to solve this for certain values of a and b, for example, if a = 0.5, and b = -1, then the probabilit...

This is the same question as:There are n steps, you can either make one or two steps a time, how many different ways to reach the top

I found a nice reference on problem like this:On the Lenghs of the Pieces of a Stick Broken at RandomBy Lars Holst, Journal of Applied Probability, Vol17, 1980

Let E be the expected toss to obtain {HT}Let x be the expected toss to obtain {HT} after HLet y be the expected toss to obtain {HT} after TE = 1 + p*x + (1-p)*yx = p(1+x) + (1-p)*1 = 1 + p*x => x = 1/(1-p)y = p*(1+x) + (1-p)*(1+y) => y = 1/p + 1/(1-p)So, E = 1/(p*(1-p))

Probability of not forming a circle is:\int_0^2\pi \phi/so, the answer is 1-0.25 = 0.75

Can somebody give a clue how to obtain the general solution, instead of putting the final result? Thanks a lot...

Thanks a lot, Jingchan.I was thinking in the same lines, but thought there might be more elegant method to solve this.

Could anyone suggest an solution to the following problem? I tried, but the algebra gets really messy...Put 18 balls into 10 box, expected number of empty boxes?