Math 10B Section
Worksheet # 16
Author: Bo Lin
Independence of random variables
Definition 1. Let X and Y be two random variables defined on the same probability space. X and Y are independent if for any subsets A, B of R, the events
X ∈ A and Y ∈ B are independent, i.e.
P (X ∈ A ∩ Y ∈ B) = P (X ∈ A) · P (Y ∈ B).
Variance, standard error and covariance
The variance measures the width of a distribution or how far a set of numbers
are spread out.
Definition 2. The variance of a random variable X is
V ar[X] = E[(X − µ)2 ],
where µ = E[X]. The standard error (standard deviation) of X is
p
σ = SE[X] = V ar[X].
The covariance of two random variables X and Y is defined as
Cov[X, Y ] = E[XY ] − E[X] · E[Y ].
Remark 3. If X and Y are independent, then Cov[X, Y ] = 0. However, the
converse is NOT true.
Theorem 4.
(i) V ar[X] = E[X 2 ] − E[X]2 ;
(ii) V ar[cX] = c2 V ar[X] for real number c;
(iii) V ar[X + Y ] = V ar[X] + V ar[Y ] + 2Cov[X, Y ];
(iv) if X1 , X2 , . . . , Xn are independent random variables, then
n
n
X
X
V ar[
Xi ] =
V ar[Xi ]
i=1
i=1
.
Remark 5. Be careful of property (iv): the same formula for expected values
applies to any random variables X1 , X2 , . . . , Xn defined on the same probability
space, no matter whether they are independent; but for variance it applies only
if X1 , X2 , . . . , Xn are independent.
1
Formula of V ar[X] for discrete X
Name
Uniform
Bernoulli
Binomial
Geometric
Hypergeometric
Poisson
p.m.f. of X
P (X = xi ) = n1 for 1 ≤ i ≤ n
P (X = 1) = p, P (X
= 0) = 1 − p
P (X = k) = nk pk (1 − p)n−k
P (X = k) = (1 − p)k p
(m)(N −m)
P (X = k) = k Nn−k
(n)
λk e−λ
P (X = k) = k!
1
n
E[X]
P
n
i=1 xi
p
np
1−p
p
1
n
V ar[X]P
n
1
2
2
x
i=1 i − n2 (
i=1 xi )
p(1 − p)
np(1 − p)
Pn
mn
N
1−p
p2
mn(N −m)(N −n)
N 2 (N −1)
λ
λ
Examples
1. Suppose you toss a fair coin once. Let X be the random variable that takes
value 2 for heads and value 0 for tails. Are X and X independent? What is
Cov(X, X)?
2. Suppose X is a binomial random variable with expected value 4 and variance
3. Then what is P (X = 2)?
Homework Review
HW11 Q5 Suppose pens are sold in packages of 20. The manufacturer guarantees that no more than one pen per package will be defective, and will replace
any package with two or more defective pens. Supposing that each pen is defective with probability 0.02, what is the probability that a package will need
to be replaced?
2