Q.la Write the pdf of a normal random variable and a standard normal random variable. If the annual rainfall in Cape Town is normally distributed with mean 20.2 inches and standard deviation 3 6 inches. Find the probability that the sum of the next five years' rainfall exceeds 110 inches. Q.1b If Cov(Xm, Xn) = mn - (m+n), find Cov(X₁ + X2, X3 + X₁).

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Q.la Write the pdf of a normal random variable and a standard normal random variable. If the annual rainfall
in Cape Town is normally distributed with mean 20.2 inches and standard deviation 3 6 inches. Find the
probability that the sum of the next five years' rainfall exceeds 110 inches.
Q.1b If Cov(Xm, Xn) = mn - (m + n), find Cov(X₁ + X2, X3 + X4).
Q.2 Starting at some fixed time, let F(n) denotes the price of a First Local Bank share at the end of n additional
weeks, n ≥ 1; and let the evolution of these prices assumes that the price ratios F(n)/F(n 1) for n ≥ 1
are independent and identically distributed lognormal random variables. Assuming this model, with lognormal
parameters μ = 0.012 and o = 0.048, what is the probability that the price of the share at the end of the four
weeks is higher than it is today?
Q.3 Explain what do you understand by a geometric Brownian motion process. Suppose that the price of a
Nedbank share follows a geometric Brownian motion described by N(y), y ≥ 0 with drift parameter μ = 0.05
and volatility parameter o= 0.39. If N(0) = 75, find E[N(6)].
Transcribed Image Text:Q.la Write the pdf of a normal random variable and a standard normal random variable. If the annual rainfall in Cape Town is normally distributed with mean 20.2 inches and standard deviation 3 6 inches. Find the probability that the sum of the next five years' rainfall exceeds 110 inches. Q.1b If Cov(Xm, Xn) = mn - (m + n), find Cov(X₁ + X2, X3 + X4). Q.2 Starting at some fixed time, let F(n) denotes the price of a First Local Bank share at the end of n additional weeks, n ≥ 1; and let the evolution of these prices assumes that the price ratios F(n)/F(n 1) for n ≥ 1 are independent and identically distributed lognormal random variables. Assuming this model, with lognormal parameters μ = 0.012 and o = 0.048, what is the probability that the price of the share at the end of the four weeks is higher than it is today? Q.3 Explain what do you understand by a geometric Brownian motion process. Suppose that the price of a Nedbank share follows a geometric Brownian motion described by N(y), y ≥ 0 with drift parameter μ = 0.05 and volatility parameter o= 0.39. If N(0) = 75, find E[N(6)].
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