Download Continuous Random Variables

Continuous Random Variables
Dennis Sun
Discrete Random Variables
So far, the random variables we’ve seen take on a few, distinct set
of values. Such variables are called discrete.
Example: If we toss a coin 5 times, the number of heads is a
discrete random variable. It only takes on the values 0, 1, 2, ..., 5.
p[x]
e.g., P (number of heads = 3) = .3125
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00 0
1
2
x
3
4
5
Discrete Random Variables
p[x]
P (number of heads > 2) is represented in red below:
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00 0
1
2
x
3
4
5
Not All Random Variables are Discrete
My bus is equally likely to be delayed by any amount of time
between 0 and 10 minutes.
There are infinitely possible values for the delay of the bus.
Therefore, the probability that delay is equal to any particular
value has to be 0,
e.g., P (delay = 6.027252) = 0.
But it still makes sense to talk about the probability that the delay
is in some range,
e.g., P (delay > 5) = .5.
Continuous Random Variables
We call random variables like the one above, that can take on any
values in a range, continuous.
p(x)
We describe continuous random variables by a probability
density function (p.d.f.), denoted p(x).
0.12
0.10
0.08
0.06
0.04
0.02
0.00 2
0
2
4
6
8
bus delay (minutes)
10 12
We can find probabilities by integrating the p.d.f.
Example
The time (in hours) it takes a randomly chosen student to finish my
final exam is a random variable with p.d.f.
(
1
(x − 1) 1 < x ≤ 3
p(x) = 2
.
0
otherwise
What is the probability that the student takes longer than 2 hours?
Expected Value
p(x)
What is the expected value of a continuous random variable?
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00 4
3
2
1 0
x
1
2
3
4
Expected Value
p(x)
What is the expected value of a continuous random variable?
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00 4
3
2
E[X] ≈
1 0
x
X
1
2
xi p(xi )∆
i
↓ ∆→0
Z
∞
E[X] =
xp(x) dx
−∞
3
4
Formulas for Continuous Random Variables
You can translate any discrete formula into a continuous one by:
• replacing the sum with an integral
• replacing the p.m.f. p[x] by p(x) dx.
Discrete
d
X
P (c ≤ X ≤ d)
p[x]
E[X]
E[g(X)]
Var[X]
Continuous
Z d
p(x) dx
c
x=c
Z
∞
X
xp(x) dx
xp[x]
−∞
x
Z
∞
X
g(x)p[x]
g(x)p(x) dx
−∞
x
Z
∞
X
(x − E[X])2 p[x]
(x − E[X])2 p(x) dx
x
−∞
Example
The time at which a randomly chosen student finishes my final
exam is a random variable with p.d.f.
(
1
(x − 1) 1 < x ≤ 3
p(x) = 2
.
0
otherwise
What is the expected time it takes for a student to finish? What is
the variance?