1. A mail-order computer business has six telephone lines. Let X ...

X = n
i=1 Xi, where each Xi ∼ BIN(1, p), (or Bernoulli with
parameter p - a r.v taking value 1 with probability p and 0 with
pr. 1 − p).
Therefore
=
E[X] = E[
=
n
Xi]
i=1
n
E[Xi] = np
i=1
= 10 × 0.5 = 5
V [X] = V [
=
n
Xi]
i=1
n
V [Xi]
i=1
n
p(1 − p) = np(1 − p)
i=1
= 10 × 0.5 × 0.5 = 2.5
Standard deviation is the square-root of variance, and hence equals
1.581139.
6. Show that E(X) = np when X is a binomial random variable. [HINT:
First express E(X) as a sum with lower limit x = 1. Then factor out
np, let y = x − 1 so that the sum is from y = 0 to y = n − 1, and show
that the sum equals 1.]
Ans
E[X] =
=
=
n
x=1
n
x=0
n
x
x
p x (1 − p) n−x
n n!
x
x!(n − x)! px (1 − p) n−x
x=0
n!
(x − 1)!(n − x)! px (1 − p) n−x
n−1 (n − 1!
= n
y!(n − y − 1)! py+1 (1 − p) n−y−1
y=0
= np(p + 1 − p) n−1 = np
5