Consider a diffusion process {Xt,t > 0} with constant drift μ(z)-a and constant volatility σ(z) >...
2. Assume X is a random variable following from N(μ, σ2), where σ > 0. (a) Write down the pdf of X (b) Compute E(X2). (b) Define YFind the distribution of Y.
4. Consider the process X+ = Vaw (t/a), where a is a positive constant. Calculate Var[X/(t+u) - X+(t)], where u > 0. Is X, a Brownian motion?
etX be normally distributed with mean μ = 120 and standard deviation σ = 20. (a) Find z such that P(X > z) = 0.90. (b) Find z such that P(80 S X 100).
Let x be an arithmetic brownian motion starting from 0 with
drift parameter 0.2
Let X-(Xt ,0 < t < 1} be an arithmetic Brownian motion starting from 0 with drift parameter μ-0.2 and variance parameter ơ2-0.125. 1. Calculate the probability that X2 is between 0.1 and 0.5 2. Given that X 0.6, find the probability that X2 is between 0.1 and 0.5 3. Given that Xi- 0.2, find the covariance between X2 and X3
IF Let x(t) Show that e 20" σ>0, and let (o) be the Fourier transform of x(t) .
Consider the random variable Z defined as follows: for z=c for z 1 2K fz(z) for zc otherwise 0 for some constant c> 1 (a) Determine the value of K. Z2 (b) Determine the distribution of the random variable W (c) Find the moment generating function of Z when c 2 (d) Using the moment generating function, determine the mean and the variance of Z when c 2
If X is a normal random variable with μ =-2 and σ = 3, and has probability density function and cumulative density function fx (z), FX (z), calculate . P(-3< X < 0) F(1/4
3. An infinite bar has initial temperature distribution: T(x,0)-T0 [s(x-l) +δ(x + 1)] Find T(x,t) for >0. The free space Green's function for the 1-D diffusion equation is G(x -x)- e 4DI 4TDt
9. IfX is a r.), distributed as N(μ, σ2), find the value of c (in terms of μ and σ) for which P(Xc 2-9P(X > c).
Let X-(Xt ,0 < t < 1} be an arithmetic Brownian motion starting from 0 with drift parameter μ-0.2 and variance parameter ơ2-0.125. 1. Calculate the probability that X2 is between 0.1 and 0.5 2. Given that X 0.6, find the probability that X2 is between 0.1 and 0.5 3. Given that Xi- 0.2, find the covariance between X2 and X3