Download Math 375 Probability Fall, 2011 Chapter 5 Outline: Continous

Math 375
Probability
Fall, 2011
Chapter 5 Outline: Continous Random Variables
September 22
Key Concepts
• If X is a continuous random variable, the probability that X lies in some region B is given by
Z
P (X ∈ B) =
f (x)dx,
B
where f (x) is the density function of X.
• The expected value of X is
Z
∞
E(X) = µX =
xf (x)dx
−∞
• The variance of X is
2
V ar(X) = σX
=
Z
∞
(x − µ)2 f (x)dx
−∞
• The standard deviation of X is the square root of the variance.
Useful Distributions
• The uniform distribution on the interval [a, b]. The density function is
1
a≤x≤b
b−a
f (x) =
0
elsewhere
The mean is (a + b)/2, the variance is (b − a)2 /12.
• The normal distribution has density function
2
2
1
f (x) = √
e−(x−µ) /σ
2πσ 2
The normal distribution is the classical “bell curve”. Its mean is µ, its variance is σ 2 . The density
function is symmetric around µ, and has 68%, 95%, and 99.9% of its area within 1, 2, and 3 standard
deviations, respectively. Probabilities for normal RVs with parameter µ = 0 and σ = 1 can be read
from a table (page 201 in the text); probabilites for normal RVs with different parameters can be
read from the same tables by first “normalizing” the RV, i.e. forming a new RV by subtracting the
mean and dividing by the standard deviation.
• The exponential distribution has density function
λe−λ x ≥ 0
f (x) =
0
elsewhere
Exponentials are often used to represent waiting times, e.g. for a bus or a tornado. The mean is
1/λ, the variance is 1/λ2 .
2
Miscellaneous Factoids
• If X is an RV with mean µ and standard deviation σ, then
Z = aX + b
has mean aµ + b and standard deviation aσ.
• As a consequence, for any random variable with mean µ and standard deviation σ,
Z = (X − µ)/σ
has mean 0 and standard deviation 1.