11.3 CONTINUOUS RANDOM VARIABLES

Objectives: (a) Understand probability density functions (b) Solve problems related to probability density function (c) Understand cumulative distribution f functions (d) Calculate the probabilities from cumulative distribution functions (e)Sketch the graph of cumulative distribution functions

A continuous random variable X is specified by its probability density function (p.d.f) which is written as f(x). This function is defined over the interval (-∞,+∞). Continuous random variables ~are theoretical representations of continuous variables such as height, weight or time.

X is known as a continuous random variable if: The probability density function (p.d.f), f(x) ≥ 0 for all x and

y = f (x) y x 0 a b NOTE 1. If X is a continuous random variable with probability density function f(x), (p.d.f), then For a and b real

2. For continuous random variable X,

Example 1 a)Show that the function is the probability density function (p.d.f) b) P(X< )

Solution: Thus, f(x) is the probability density function (p.d.f)

b) P (X < )

A continuous random variable X has probability density function Example 2 Find a)the value of the constant k. b)P(0.3 ≤ X ≤ 0.6)

Solution: a)

b) P(0.3 ≤ X ≤0.6)

A continuous random variable X has probability density function Example 3 Find a) the value of the constant k. b) sketch the probability density function c) find P(1.5 ≤ X ≤ 2.5) d) find P(X > 1.8)

k(4 – 0) + 2k(3-2) = 1 Solution: a) 6 k = 1

Solution: b) y x 0 312

= c) P(1.5 ≤ X ≤ 2.5)

P(X>1.8) d) = 1 - P( X ≤ 1.8) = 0.43

If X is a continuous random variable with probability density function f(x) for -∞<x<∞, then the cumulative distribution function (c.d.f.) of X is given by

Notes 1.F(x) is in fact given by the area under the curve f(x) from -∞ up to x as indicated by the shaded region F(x) y x 0 x y=f(x)

Example 3 X is a continuous random variable with probability density function a) Find cumulative distribution function F(x) b) Sketch the graph F(x) c) Calculate P(2<X<3)

Solution: For x<0:a)

Therefore, the cumulative distribution function is

b) y x 0 4 1

= F(3) c) P(2 < X < 3) - F(2)

then X is known as a continuous random variable. Theorem If the probability density function f(x) ≥ 0 for all x and