Download Oct. 4 3.3 Continuous-Type Random Variables Oct. 4 3.3

Oct. 4
3.3
Continuous-Type Random Variables
Recall what discrete means: Support is a finite or countably infinite set.
Then, we added using a mass function.
Now, we will integrate using a density function.
Then, a single point could have positive probability mass.
Now, only intervals of the real line have positive mass.
Oct. 4
3.3
Continuous-Type Random Variables
Properties of the probability density function (p.d.f.) of a continuous r.v.:
(Compare to the p.m.f. properties on page 60.)
f (x) ≥ 0 for all real numbers x.
Z ∞
f (x) dx = 1.
−∞
For any real interval (a, b),
Z
P(a < X < b)P(a ≤ X < b)P(a < X ≤ b)P(a ≤ X ≤ b) =
b
f (x) dx.
a
Oct. 4
3.3
Continuous-Type Random Variables
Always pay attention to the support of a continuous r.v.!
Example (pages 142–143): Suppose X has density
(
1 −x/20
e
if 0 ≤ x < ∞,
fX (x) = 20
0
otherwise.
Z
Then
∞
Z
fX (x) dx
−∞
becomes
0
∞
∞
1 −x/20
e
dx = −e−x/20 = 1.
20
0
Oct. 4
3.3
Continuous-Type Random Variables
Find c if Y has density function
f (y ) = cy (2 − y ) for 0 < y < 2.
Oct. 4
3.3
Continuous-Type Random Variables
Then, we added using a mass function.
Now, we will integrate using a density function.
X
For discrete X , we learned that E(X ) equals
xf (x).
x∈S
Z
∞
For continuous X , E(X ) =
xf (x) dx.
−∞
Oct. 4
3.3
Continuous-Type Random Variables
Other formulas for discrete random variables carry over:
Var (X ) = E[(X − µ)2 ] = E(X 2 ) − µ2 . (And the std. dev. is
MX (t) = E etX
√
σ 2 .)
However, these things require a nontrivial fact:
Z ∞
E[u(X )] =
u(x)f (x) dx.
−∞
Oct. 4
3.3
Continuous-Type Random Variables
The (cumulative) distribution function is important for a continuous r.v.
As before, F (x) = P(X ≤ x).
Z x
Thus, F (x) =
f (t) dt.
−∞
But now, we also have f (x) = F 0 (x).
Incidentally, the c.d.f. of a continuous r.v. is — you guessed it —
continuous! (In contrast to the discrete case.)
Oct. 4
3.3
Continuous-Type Random Variables
Note: A c.d.f. always satisfies
lim F (x) = 0 and lim F (x) = 1.
x→∞
x→−∞
Examples:
Find F (x) if f (x) =
1 −x/20
20 e
for x > 0.
Find F (y ) if Y has density function
f (y ) = cy (2 − y ) for 0 < y < 2.
Oct. 4
Desired Outcomes
Students will be able to:
list the properties of a density function for a continuous r.v.;
find the constant multiplier for a density by setting the integral to 1;
give formulas for mean and variance of a continuous r.v.;
differentiate the c.d.f. of a continuous r.v. to find a p.d.f.;
integrate a p.d.f. to find the c.d.f.