FinMath September Review 2013
Exercises for Greg Lawler’s second lecture
Exercise 1. Suppose that X has a uniform distribution on the interval [0, 2].
• Give the distribution function, density, mean, and variance of X.
• Find the moment generating function and characteristic function of X.
• Let Y = X 3 . Find the density of Y .
• Suppose that X1 , X2 are independent each with the uniform distribution on [0, 2]. True
or false: X1 + X2 has a uniform distribution on [0, 4].
Exercise 2. Do a computer simulation of the following. Suppose one is catching fish on a
Saturday morning. One starts at 8:00 and stops at 12:00. The waiting time between fish
caught is exponential with parameter λ = .9 (time is measure in hours). Do many simulations
to see how many fish are caught by noon in order to get an estimate for
p(k) = P{exactly k fish caught in four hours}.
Compare your simulation to a Poisson random variable with parameter 3.6. (Why are we
choosing parameter 3.6?)
Exercise 3. Suppose X1 , X2 , X3 are independent random variables each normal with mean
zero and variances 1, 4, 9, respectively. Find the following probabilities to four decimal places.
• P{X1 /X3 > 0}
• P{X3 > X1 + X2 + 1}
• P {X12 + (X2 /2)2 + (X3 /3)2 ≤ 11.35} .
Exercise 4. Suppose X and Y are independent random variables with exponential distributions with parameters λ1 and λ2 . Let Z = min{X, Y }, W = max{X, Y }.
• Show that Z has an exponential distribution. What is the parameter?
• Does W has an exponential distribution? Why or why not?
• Suppse λ1 = λ2 ? Does W − Z have an exponential distribution?
• How about if λ1 6= λ2 ?
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