Use Ito's formula (1.15) to write the below stochastic processes Y(t) in the form dY(t) = Udt+VdB(t) (a) Y(t) = B²(t) (b) Y(t) = 2+t+ €³(t).
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- A random process x(t) is applied to a network with impulse response h(t) u(t) t exp (- bt) where b>0 is a constant. The cross-correlation of x(t) with the output y(t) is known to have the same form. Rxx(t) = u(t) T exp (– bt) (a) Find the auto correlation of y(t) ? (b) What is the average power in y(t) ?A Random process X(t) is applied to a network with impulse response h(t) = u(t) exp (-bt) where b > 0 is a constant. The cross- correlation of X(t) with the output Y(t) is known to have the same form, Ryy(t) = u(t)t exp(-bt) (i) Find the auto-correlation of Y(t).(b) Let W(t) be Brownian motion. Using Ito's lemma evaluate | (4w*(t) – 121W (t))dW(t) (c) St follows a stochastic process given by dS; = (a – bS;)dt + odW; where Wt is a Brownian motion and a, b, o are all positive. The process Y; is given by Y = exp(bt)St. Calculate dYt.
- (4) Find β such that the equation βx2+ x + β = 0 has a repeated real solution.2. Let Y = aX + b %3D (1) Find the covariance of X and YSuppose that {W(t)};20 is a standard Brownian motion with associated filtration {F(t)}t>0. Then the stochastic process W(t)? - t is a martingale with respect to its associated filtration {F(t)}>0- TRUE/FALSE
- Consider the stochastic process Y, = (1+y)B1 + (1+ e)(B? – t), where Bị is a Brownian motion. Show that it is a Martingale.5 The relation between the input X (t) and Y (t) of a system is Y (t) = X² (t). X (t) is a zero mean stationary Gaussian random process with auto correlation function (T) = e- RxX = e- a ld for a > 0. Find E[Y (t)] and Ryy (t).Example 21: A random process is given by Z (t) = AX (t) + BY (t) where A and B are real constants and X (t) and Y (t) are jointly WSS processes. Find (i) dzz (@) %3D
- 5- Suppose that { W(t): t 0} is a Brownian motion process with variance ow = 3. Find P{ W(1) < 1| W(2) = 5/4 } = ?Consider a logistic regression system with two features x1 and x2. Suppose 00 = 5, 01 = 0, 02= 0, 03= -5, 04= -1, draw the decision boundary of hø(x) = g(0+ 01 x1+ 02 X2 + 03 xı²+ 04 x2?).b) Show that Δ2y0 = y2-2y1 + y0.