Write a python 3 code for the following parts:

(a) 13 points] Generate a sample path of Poisson process {N(t)0 with intensity 10 Plot the sample path of N(.) for the time interval [0, 10 (b) 13 points] Investors purchase $1000 bonds at the random times of a Poisson process N(t) 0 with parameter 10 . With the interest rate r = 0.03 (constant), the present value (that is, at time 0 value) of an investment purchased at time t is given by 1000et. The present value X(t) of the bonds purchased by time t is defined by N (t) i=1 where Wi, W2, .. are the event times of N(). Generate a sample path of X(t), t20. c[4 points] Estimate E[X (5)] by simulating X () many times. Estimate Var(X(5)). Com pare with the theoretical values (see (9). (a) 13 points] Generate a sample path of Poisson process {N(t)0 with intensity 10 Plot the sample path of N(.) for the time interval [0, 10 (b) 13 points] Investors purchase $1000 bonds at the random times of a Poisson process N(t) 0 with parameter 10 . With the interest rate r = 0.03 (constant), the present value (that is, at time 0 value) of an investment purchased at time t is given by 1000et. The present value X(t) of the bonds purchased by time t is defined by N (t) i=1 where Wi, W2, .. are the event times of N(). Generate a sample path of X(t), t20. c[4 points] Estimate E[X (5)] by simulating X () many times. Estimate Var(X(5)). Com pare with the theoretical values (see (9)