Chapter 6
Continuous random variables
• probability density function (f (x))
the probability distribution function of a continuous random variable (analogous to the probability mass function for a discrete random variable), which takes only
positive values at any possible value of the variable, and
such that the total area under its graph equals 1;
P (a ≤ X ≤ b) = the area under the graph of f (x)
over the interval a ≤ X ≤ b
• cumulative distribution function (F (x))
the function equal to the area under the graph of f (x)
over the interval X ≤ x; it follows that
P (a ≤ X ≤ b) = F (b) − F (a)
1
Chapter 6
Uniform random variables
• uniform distribution
assigns probability uniformly across all values of a continuous random variable X, that is, any interval of values of X of a given size is assigned the same probability:
for any numbers r, s, t for which the following equation
makes sense, P (s ≤ X ≤ t) = P (r + s ≤ X ≤ r + t).
• uniform density function
Where X takes on values in the interval a ≤ x ≤ b,
(
1
if a ≤ x ≤ b
b−a
f (x) =
0
if x < aor x > b
• uniform distribution parameters
if X is a uniform random variable, then
a+b
E(X) = µ =
2
(b − a)2
2
V ar(X) = σ =
r 12
(b − a)2
SD(X) = σ =
12
2
Chapter 6
Normal random variables
• normal distribution
– by far the most common probability distribution in
statistics, and central to the underlying theory
– characterized by the iconic bell-shaped curve
– completely determined by knowledge of two parameters, its mean µ and standard deviation σ
– symmetric about its mean value
– the region lying within one standard deviation of the
mean is the interval where the bell curve is concave
down
– a normal variable X can take any real number value
(−∞ ≤ X ≤ ∞), but X is less likely to take values
further away from the mean, i.e., the distribution
is asymptotic (meaning that in either direction the
tails of the distribution curve approach – without ever
touching – the horizontal axis)
3
Chapter 6
• normal density function
Where X has mean value µ and standard deviation σ,
1
−(x−µ)2 /2σ 2
f (x) = √ e
σ 2π
• normal distribution parameters
if X is a normal random variable, then
E(X) = µ
V ar(X) = σ 2
SD(X) = σ
4
Chapter 6
The standard normal random variable
Any normal random variable X with mean µ and standard
deviation σ can be transformed into the standard normal
random variable Z by means of the rescaling formula
X −µ
.
Z=
σ
Whenever X takes a value x, Z will take the corresponding
standardized value
x−µ
z=
.
σ
By construction, Z always has mean value 0 and standard
deviation 1.
The Empirical Rule
The percentages listed in the Empirical Rule come from
computing the normal probabilities:
P (µ − σ ≤ X ≤ µ + σ) = 0.6826
P (µ − 2σ ≤ X ≤ µ + 2σ) = 0.9544
P (µ − 3σ ≤ X ≤ µ + 3σ) = 0.9972
5
Chapter 6
Exponential random variables
• exponential random variable
measures the time X between Successes in a Poisson
process, a process in which events occur continuously
and independently at a fixed rate of λ Successes per
unit time (λ is called the rate parameter)
• exponential probability density function
since we must have X ≥ 0, the density function is
undefined for negative values; for x ≥ 0,
f (x) = λe−λx
• exponential cumulative density function
where X ≥ 0 is an exponential random variable,
F (x) = P (X ≤ x) = 1 − e−λx
• exponential distribution parameters
if X is an exponential random variable, then
1
1
and SD(X) = σ =
E(X) = µ =
λ
λ
6
Chapter 6
Lognormal random variables
• lognormal random variable
a random variable Y whose logarithm X = ln Y is
normally distributed; useful for describing some positive
variable quantities with positive skew: incomes, prices,
times between Successes in situations where the rate of
failure is not constant over time, etc.
• lognormal probability density function
where Y is lognormal, with X = ln Y normal having
mean µ and standard deviation σ, then for any y > 0,
1
2
2
f (y) = √ e−(ln y−µ) /2σ ;
yσ 2π
therefore, the relationship
P (a ≤ Y ≤ b) = P ( ln a ≤ X ≤ ln b )
allows lognormal probabilities for Y to be computed
using normal probabilities for X
7
Chapter 6
• lognormal parameters
where Y is lognormal (i.e., X = ln Y is normal having
mean µ and standard deviation σ), then Y = eX and
µ+ 21 σ 2
E(Y ) = µY = e
q
SD(Y ) = σY = (eσ2 − 1)e2µ+σ2
where
µ = ln
!
µ2Y
p
µ2Y + σY2
8
s σY2
and σ = ln 1 + 2
µY