HW 5

Please answer all part : Let rt be a log return. Suppose that r0, r1, . . . are i.i.d. N(0, 0.01^2). (a) What is the distribution of rt(8) = rt + rt−1 + rt−2 +...+ rt−7? (b) What is the covariance between r7(3) and r9(3)? (c) What is the conditional distribution r17(3) given that r16 =0.004 (d) What is the probability that the gross return over the first 10 times periods is at least 1.05?

Pb(Z = z) = B (8.25z - z2) with z E {1, 2, ......., 5}.   What value must B be to ensure the probability mass function for Z, equation(1), is valid? I asked this question before but the experts writting was difficult to read. Can you please answer with clear writting. Thanks.

D6) Finance Suppose that the stock price follows geometric Brownian motion dXt = 0.04 Xt dt + 0.2 Xt dWt. Write the probability density function for the distribution of the stock prices in 2 years if the current stock price is 80.

EXER 6.3 Find the covariance and the correlation coefficient between X and Y, if X and Y are jointly discrete random variables, with joint PMF given by: SHOW SOLUTIONS X\Y 0 1 6 0 28 6 1 28 2 0 333333 28 28 28 2120 28 0

Q.3 The probability density function for a continuous random variable X is fx(x) = }" (a+ bx2; 0 < x< 1, 0; %3D e.w. (i) Find constants a and b if E (X) = 0.6 (ii) Find cumulative distribution function (cdf), moment generating function. (iii) Using mgf, find ß, and ß2 and comment on it. (iv) Sketch pdf and cdf. %3D

EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.