Probability Lecture 10: Continuous Random Variables, Study notes of Statistics

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STAT 430/510 Probability

Hui Nie

Lecture 10

June 15th, 2009

Introduction

The set of possible values for discrete random variable is either finite or countably infinite However, there also exist random variables whose set of possible values is uncountable.

Properties of Continuous Random Variable

1 = P(X ∈ (−∞, ∞)) =

−∞ f^ (x)dx P(a ≤ X ≤ b) =

∫ (^) b a f^ (x)dx P(X = a) =

∫ (^) a a f^ (x)dx^ =^0 P(X < a) = P(X ≤ a) =

∫ (^) a ∞ f^ (x)dx

Example

The amount of time in hours that a computer functions before breaking down is a continuous random variable with probability density function given by

f (x) =

λe−x/^100 , x ≥ 0 0 , x < 0

What is the probability that (a) a computer will function between 50 and 150 hours before breaking down? (b) it will function for fewer than 100 hours?

Example

If X is continuous with distribution function FX and density function fX , find the density function of Y = 2 X.

Determine FY by

FY (a) = P(Y ≤ a) = P( 2 X ≤ a) = P(X ≤ a/ 2 ) = Fx (a/ 2 )

Differentiating FY gives fY (a) = 12 fX (a/ 2 )

Proposition 5.2.

If X is a continuous random variable with probability density function f (x), then, for any real-valued function g,

E[g(X )] =

−∞

g(x)f (x)dx

Properties of Expected Value and Variance

E[aX + b] = aE[X ] + b, where a and b are constants Var (aX + b) = a^2 Var (X ) , where a and b are constants

Example

Find E[X ] and Var(X) when the density function X is

f (x) =

2 x, 0 ≤ x ≤ 1 0 , otherwise

E[X ] =

0 x^2 xdx^ =^2 /^3 E[X 2 ] =

0 x

(^22) xdx = 1 / 2 Var (X ) = E[X 2 ] − (E[X ])^2 = 1 / 18

Example

A stick of length 1 is split at a point U that is uniformly distributed over (0,1). Determine the expected length of the piece that contains the point p, 0 ≤ p ≤ 1.

Example

Suppose that if you are s minutes early for an appointment, then you incur the cost cs, and if you are s minutes late, then you incur the cost ks. Suppose also that the travel time from where you presently are to the location of your appointment is a continuous random variable having probability density function f. Determine the time at which you should depart if you want to minimize your expected cost.