Recent content by fignewtons

Homework Statement Imagine you are playing a game with me, of drawing balls from a box. There are two blue balls and two red balls. They are picked with equal probability, and are drawn without replacement. If you draw a blue ball, I give you $1. If you draw a red ball, you pay me $1.25. What...

Homework Statement I want to generate two random variables, one is normally distributed N ~N(10, 25) and the other one, E, is exponentially distributed with mean 1. I was not given a particular correlation coefficient.Homework Equations normal cdf, exponential cdf, inverse transform method...

EDIT_I noticed that I copied down the equation for C incorrectly because I looked up the incorrect Z_3,1. After the revision, I did get C~N(0,1). Thanks for the help! ------------------------- The Z matrix I got was $$Z = \begin{bmatrix} 1 & 0 & 0 \\ .3 & \sqrt{.91} & 0 \\...

Homework Statement Given correlation matrix $$M = \begin{bmatrix} 1 & .3 & .5 \\ .3 & 1 & .2 \\ .5 & .2 & 1 \\ \end{bmatrix}$$ And 3 independent standard normals $$N_1, N_2, N_3$$ using cholesky decomposition A) get the correlated standard normals B) and if...

Ok I see what you're saying. But I'm still not sure how to write S appropriately given that the definition is only for a walk starting at 1. I just abandoned the notation instead for a more explicit and general one. Let Sa,b = min{t>0, Wt(a) = b} I get E[S1,0|R(1)] = E[S1,0 | X1 = +1 ]P[X1 =...

Ok I thought i was going to make the notation easier but perhaps I made it more confusing so I'm just going to quote verbatim from the problem. Given 1 > p > 1/2 > q=1-p > 0, let R(a) be the event that the random walk Wt(a) starting at a goes back to 0. R(a) = {Wt(a) = 0 for some t > 0, a > 0}...

Ok, so we can take Δt again to be a small number, just that t=0. And as Δt -> 0, P[B ring] -> 0

thanks for the hint. i think the quitting criterion tells you to quit or continue at end of Δt given you start at t. we want max Δt because playing longer allows for chance of winning more dollars, so the least t can be is 0 (which allows for the max Δt). In this case, t=0. so r(1-0)=r...the...

Homework Statement Let w(1) = event of a random walk with right drift (p > q, p+q = 1) starting at 1 returns to 0 Let p(w(1)) = probability of w(1) Let S=min{t>=0:wt(1)=0} be the minimum number of steps t a walk starting from 1 hits 0. What is E[S|w(1)]? Homework Equations I know E[S|w(0)] = 0...

Sorry for being thick, but I am confused, is the maximal winning r(1-t) as in the quitting criterion verbatim? or is it r(1-t) -1 since x<r(1-t) means it can never reach r(1-t) exactly, and since x is discrete and increases only in increments of $1. or am I completely off?

So the maximal x is just r(1-t)? If you decide to stop at any time t.

Homework Statement You are playing a game with two bells. Bell A rings according to a homogeneous poisson process at a rate r per hour and Bell B rings once at a time T that is uniformly distributed from 0 to 1 hr (inclusive). You get $1 each time A rings and can quit anytime but if B rings...

Homework Statement I have n balls and m urns numbered 1 to m. Each ball is placed randomly and independently into one of the urns. Let Xi be the number of balls in urn number i. So X1+...+Xm = n What is the distribution of each Xi? What is EXi and VarXi What is E[XiXj] given i≠j What is...

Ok fixed it. Step 1: let X0 = 1 if U0 ≤ ν1 let X0 = 2 if ν1 < U0 ≤ ν1 + ν2 ... more generally let X0 = s if ν1 + ν2 + ... + νs-1 < U0 ≤ ν1 + ν2 + ... + νs-1 + νs set n = 1 Step 2: If Xn-1 = 1 let Xn = 1 if Un ≤ P11 let Xn=2 if P11 < Un ≤ P11 + P12 let Xn=3 if P11 + P12 < Un ≤ P11 + P12 + P13...

Ok after some more research I tried. Step 2: If Xn-1 = 1 let Xn = 1 if Un ≤ P11 let Xn=2 if P11 < Un ≤ P11 + P12 let Xn=3 if P11 + P12 < Un ≤ P11 + P12 + P13 ... let Xn=s if P11 + P12 + ... + P1(s-1) < Un ≤ 1 ...more generally If Xn-1 = s let Xn = 1 if Un ≤ Ps1 let Xn = 2 if Ps1 < Un ≤ Ps1 +...