Part V: Continuous Random Variables
http://rchsbowman.wordpress.com/2009/11/29
/statistics-notes-%E2%80%93-properties-of-normal-distribution-2/
Chapter 24: Probability Density Functions
http://divisbyzero.com/2009/12/02
/an-applet-illustrating-a-continuous-nowhere-differentiable-function//
Comparison of Discrete vs. Continuous
(Examples)
Discrete
Continuous
Counting: defects, hits, die Lifetimes, waiting times,
values, coin heads/tails,
height, weight, length,
people, card
proportions, areas,
arrangements, trials until
volumes, physical
success, etc.
quantities, etc.
Comparison of mass vs. density
Mass (probability
mass function, PMF)
0 ≤ pX(x) ≤ 1
Density (probability density
function, PDF)
0 ≤ fX(x)
∞
𝑝𝑋 𝑥 = 1
𝑥
𝑓𝑋 𝑥 𝑑𝑥 = 1
−∞
P(0 ≤ X ≤ 2) = P(X = 0)
𝑃
0
≤
𝑋
≤
2
=
+ P(X = 1) + P(X = 2)
P(X ≤ 3) ≠ P(X < 3)
when P(X = 3) ≠ 0
2
𝑓𝑋 𝑥 𝑑𝑥
0
P(X ≤ 3) = P(X < 3)
since P(X = 3) = 0 always
Example 1 (class)
Let x be a continuous random variable with
density:
1
𝑓𝑋 𝑥 = 24 2𝑥 + 3 1 ≤ 𝑥 ≤ 4
0
𝑒𝑙𝑠𝑒
a) What is P(0 ≤ X ≤ 3)?
b) Determine the CDF.
c) Graph the density.
d) Graph the CDF.
e) Using the CDF, calculate
P(0 ≤ X ≤ 3), P(2.5 ≤ X ≤ 3)
Example 1 (cont.)
1
f(x)
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5 1
x
0.8
F(x)
-1
0.6
0.4
0.2
0
-1
0
1
2
x
3
4
5
Example 2
Let X be a continuous function with CDF as
follows
0
𝑥<0
𝐹𝑋 𝑥 = 𝑥 2 0 ≤ 𝑥 ≤ 1
1
1<𝑥
What is the density?
Comparison of CDFs
Function
Discrete
Continuous
𝐹𝑋 𝑎 = 𝑃 𝑋 ≤ 𝑎
𝐹𝑋 𝑎 = 𝑃 𝑋 ≤ 𝑎
=
=
𝑃(𝑋 = 𝑎)
𝑥≤𝑎
graph
graph
Step function with
jumps of the same
size as the mass
Range: 0 ≤ X ≤ 1
𝑎
𝑓𝑋 𝑥 𝑑𝑥
−∞
continuous
Range: 0 ≤ X ≤ 1
Example 3
Suppose a random variable X has a density given
by:
4
𝑘𝑥
0
<
𝑥
<
4
𝑓𝑋 𝑥 =
0
𝑒𝑙𝑠𝑒
Find the constant k so that this function is a
valid density.
Example 4
Suppose a random variable X has the following
density:
1
0<𝑥<1
2
𝑓𝑋 𝑥 = 1
1≤𝑥≤4
6
0
𝑒𝑙𝑠𝑒
a) Find the CDF.
b) Graph the density.
c) Graph the CDF.
-1
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
x
1
0.8
F(x)
f(x)
Example 4 (cont.)
0.6
0.4
0.2
0
-1
0
1
2
x
3
4
5
Mixed R.V. – CDF
Let X denote a number selected at random from
the interval (0,4), and let Y = min(X,3).
Obtain the CDF of the random variable Y.
1
0.8
0.6
0.4
0.2
0
-1
0
1
2
3
4
Example: percentile
Let x be a continuous random variable with
density:
1
𝑓𝑋 𝑥 = 24 2𝑥 + 3 1 ≤ 𝑥 ≤ 4
0
𝑒𝑙𝑠𝑒
0
𝑥<1
1 2
𝐹𝑋 𝑥 =
𝑥 + 3𝑥 − 4 1 ≤ 𝑥 ≤ 4
24
1
4<𝑥
a) What is the 99th percentile?
b) What is the median?