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英文数学作业代写 Math 135A代写

2021-10-26 12:15 星期二 所属: 作业代写 浏览:34

英文数学作业代写

Math 135A: HW4

Due Friday, February 26

英文数学作业代写 Consider a random walk  on the integer lattice  Z.  At  each  unit time step the walkertakes  one step  to the left or one step to the right.

1.Let X1, X2, and X3denote continuous random variables with joint density function fX1,X2 ,X3 . That is   英文数学作业代写

Defifine

Y1 := X1, Y2 := X1X2, Y3 = X1X2X3.

a)Find the joint density function for Y1, Y2, andY3.

b)AssumeX1X2, and X3 are independent and uniformly distributed on [0, 1]. Find the densityfunction for Y3. Verify that your result implies E(Y3) = 1/8 (why is this easy to see without computing the density function of Y3?)

英文数学作业代写
英文数学作业代写

2.Consider a random walk  on the integer lattice  Z. 英文数学作业代写

At  each  unit time step the walkertakes  one step  to the left or one step to the right. If the walker is at  an even  numbered lattice  site,  the probability  of one step to the right is pe and one step to the left is qe := 1 − pe. Similarly, if the walker is at an  odd numbered lattice site,  the probability  of one step  to the right  is po  and one step  to the left  is  qo := 1 − po.  Let T e  denote the time for the first passage to the site +1 given  that the walker  startsat the side  Let T o denote the first passage time to the site +2 given that the walker starts at the site +1.

Introduce the generating functions

a)Show that F eand F o satisfy the equations

英文数学作业代写
英文数学作业代写

b)Fromthe above equations derive a quadratic equation for F e(s). Solve your equation for F e(s).I got the result

Note the algebraic identity peqo − poqe pe − po. Note that when pe po p the above result reduces to  英文数学作业代写

which was derived in class.

3.Arandom walk in discrete time is performed on the graph shown in Figure  From each vertex the walker is equally likely to choose one of the neighbors connected by a bond. 英文数学作业代写

Using the labelling in the figure, write down the transition matrix P . Find the invariant measure π; that is, the probability measure that satisfies π · P π.

英文数学作业代写
英文数学作业代写

4.The cumulant generating function KX(s) of the random variable X is defined by 文数学作业代写

KX (s) = log(E(esX))

Assuming that KX has a convergent Taylor expansion

the κn(X) is called the nth cummulant of X.

a)Express κ1(X), κ2(X), and κ3(X) in terms of the moments of X.

b)Let X be normally distributed with mean zero and variance 1, find the cumulants of X. Hint: First find an expression for E(sX) and then take the logarithm.

c)Suppose X and Y are independent random variables. Showthat

κn(X + Y ) = κn(X) + κn(Y ).

英文数学作业代写
英文数学作业代写

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