Question

For this question, you may (where appropriate) write down an Excel formula using one of the functions listed below instead of providing a numerical answer. However, you must specify clearly the value of each parameter in the Excel function (you may use the 2010 or the 2007 version). If you choose to provide a numerical answer, it should be accurate to 3 significant figures.

(a) Let F denote the cumulative distribution function (cdf) of a uniformly distributed random variable X. If F(2) = 0.3, what is the probability that X is greater than 2 ?

(b) Let F denote the cdf of a uniformly distributed random variable X. If F(2) = 0.3, and F(3) = 0.6, what is F(6) ?

(c) X is a normally distributed random variable with mean 5 and standard deviation 2. What is the probability that X = 6 ?

(d) Suppose X and Y are independent random variables with standard deviations 3 and 2 respectively. Define Z = X − 2Y + 3. What is the standard deviation of Z?

(e) Suppose X and Y are Poisson random variables. X has a mean of 1 and Y has a mean of 2. X and Y are correlated with CORR (X,Y) = 0.5. What is the variance of X + Y ?

Answer 1

a) It is given that . Now the probability,

b) Let the Uniform random variable be . Then

Solving together,

Now,   since

c) The point probability of continuous distribution is 0. Hence

d) Here . The variance is

The standard deviation is

e) Variances of Poisson random variables are equal to means.

The variance of