Download probability, random variables etc

Revision material on probability
Stock market prices are unpredictable and are modelled with random walks (or Brownian
motion for continuous price models). The following material from Core A Probability will
be assumed. Look in your course notes or a book if you need more detail. Higham is useful
(but will not be followed closely) and relevant chapters will be indicated by H.x.
H.3
Events and random variables: events are subsets of the sample space Ω and every event
A ⊂ Ω has a probability P(A), where P obeys the standard axioms. A random variable X,
say, is a mapping from Ω to R and we denote events by expressions like X ≥ 3, 1 < X < 5,
(X1 = 3, X2 ≤ 4) which are abbreviations for {ω ∈ Ω : X(ω) ≥ 3} etc.
Random variables are called discrete when they map to a countable subset of R. The
function p(x) = P(X = x) (defined for all real x) is called the probability density function
of X.PThe probability of any event described using X can be written as a sum e.g. P(X ≤
3) = x≤3 p(x) so the density of X determines its probability distribution.
The binomial distribution Bin(n, p) is associated with the (random) number of successes from a sequence of n independent trials each with success probability p. If
X ∼ Bin(n, p) then
n k
p (1 − p)n−k ,
k = 0, 1, . . . , n
P(X = k) =
k
We also consider continuous random variables X. We will suppose there exists a nonnegative function p such that, for any subinterval (a, b) of R,
Z b
Z ∞
P(a < X < b) =
p(x)dx
and in particular
p(x)dx = 1.
a
−∞
and again we call p the probability density function (or pdf ) of X.
The Normal distribution is particularly important. Let µ, σ be real numbers with
σ > 0. We say that a random variable X has a Normal distribution with parameters
µ, σ 2 or X ∼ N(µ, σ 2 ) when the pdf of X is
2
1
1 x−µ
p(x) = √
,
−∞ < x < ∞
exp −
2
σ
σ 2π
2
The case where µ = 0, σ = 1 is special and we say a rv Z with pdf φ(z) = √12π e−z /2
for −∞ < z < ∞R has the standard Normal distribution. Its cumulative distribution
z
function Φ(z) = −∞ φ(s) ds is tabulated. We will reserve the notation φ(z) and Φ(z)
for these functions. For any constants µ, σ we have σZ + µ ∼ N(µ, σ 2 ).
Independence Any collection of random variables (X1 , . . .Q, Xn ) on Ω has a joint density
p(x1 , . . . , xn ). When p can be factorised i.e. p(x1 , . . . , xn ) = nj=1 pj (xj ), where each pj (·) is
a density function, then the rvs X1 to Xn are independent.
Expectation For any random variable X with density function p the expected value (or
mean) of X is the number
( P
X discrete
x xi p(xi )
E(X) = R ∞ i
xp(x) dx X continuous
−∞
as long as the sum/integral exists. The expectation of X summarises the ‘long-run average’
behaviour of X (see LLN below). Two useful measures of how much we expect X to vary
from E(X) are the variance and standard deviation, defined by
p
2
Var(X) = E [X − E(X)]
and SD(X) = Var(X)
In particular if X ∼ Bin(n, p) then E(X) = np, Var(X) = np(1 − p) while if X ∼
N(µ, σ 2 ) then E(X) = µ, Var(X) = σ 2 .
Expectation is basically integration so it is additive and in particular if X1 , X2 , . . . , Xn
are jointly distributed and they all have finite expectations then
E
n
X
Xi =
i=1
n
X
E(Xi )
i=1
The situation for variance is more complicated. For any finite collection of rvs X1 , . . . , Xm
with finite variances we have
Var
m
X
i=1
Xi =
m
X
Var(Xi ) + 2
i=1
X
Cov(Xi , Xj )
i<j
where Cov(Xi , Xj ) = E(Xi Xj ) − E(Xi )E(Xj ) (so Xi , Xj independent ⇒ Cov(Xi , Xj ) = 0).
Limit theorems For any sequence of independent
Pn random variables, X1 , X2 , . . . with
1
2
E(Xi ) = µ and Var(Xi ) = σ for all i, let X n = n i=1 Xi denote the average of n of these
rvs. The law of large numbers says that for any t > 0
σ 2 /t2
σ2
−→ 0 as n → ∞
P |X n − µ| ≥ t ≤ 2 =
nt
n
(LLN)
i.e. the average rv has a very high probability of being very near the expected value µ when
n is large.
√
Now set Zn = (X n − µ)/(σ/ n). The central limit theorem says
P(Zn ≤ c) → Φ(c) as n → ∞
(CLT)
for any number c i.e. the probability distribution of the sample average X n is approximately
N(µ, σ 2 /n) for large enough n.
Standard Normal probabilities
0.1
0.2
0.3
0.4
0.5
z
Φ(z) 0.540 0.580 0.618 0.655 0.691
z
1.1
1.2
1.3
1.4
1.5
Φ(z) 0.864 0.885 0.903 0.919 0.933
z
2.1
2.2
2.3
2.4
2.5
Φ(z) 0.982 0.986 0.989 0.992 0.994
Linear interpolation is accurate to 3 decimal places;
for z > 3, P (Z < z) ≈ 1 − φ(z)/z is adequate (where
0.6
0.7
0.8
0.9
1.0
0.726 0.758 0.788 0.816 0.841
1.6
1.7
1.8
1.9
2.0
0.945 0.955 0.964 0.971 0.977
2.6
2.7
2.8
2.9
3.0
0.995 0.996 0.997 0.998 0.999
φ(z) = (2π)−1/2 exp(− 12 z 2 ) ).