. Let X be a continuous random variable with density f, expectation E[X] = µ and variance Var[X] = σ2. Define a new random variable Y := aX + b for some a, b ∈R.
a) Calculate the standard deviation SD[Y ].
b) Express the moment generating function mY in terms of mX.
2. Prove that for an arbitrary continuous random variable X with density f we have
3. Assume that U0,1 is a uniformly distributed random variable on the unit interval. Find a real-valued function g : [0,1) →R such that Y := g(U0,1) is an exponentially distributed random variable with parameter λ > 0.
Note: This approach is very important for the simulation of distributions using a computer. The built-in (pseudo-)random number generator produces standard uniform distributed random variables. The transformation in thi example shows how the random number generator can be used to generated random numbers following an exponential distribution. This can be generalized to other distribution and is often referred to as ”inverse transform sampling”.
2
4. The lifetime of an electrical device (in months) is given by the continuous random variable X with density
if x > 0;
( ) =
0 if x ≤ 0.
a) What is c?
b) What is the probability that the device functions more than 5 months?
c) What is the expected lifetime of the device?
5. Assume that X is an exponentially distributed random variable with parameter λ > 1. Calculate
a) E[X3];
b) E[eX].
Why did we impose the condition λ > 1 (instead of the “usual” one, λ > 0)?
6. Find the cumulative distribution function F such that it has hazard rate (for t > 0). Can you express F in terms of an exponentially distributed random variable?
8 points per problems
Additional practice problems (purely voluntary - no points, no credit, no grading):
Standard Carlton and Devore, Section 3.1: Exercises 12, 17; Section 3.2: Exercises 25, 26, 34, 37; Section 3.4: Exercises 71, 73, 79, 82;
