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

15. A random sample of n bike helmets manufactured by a certain company
is selected. Let X = the number among the n that are flawed
and let p = P (flawed). Assume that only X is observed, rather than
the sequence of Successes and Failures. Derive the Maximum Likelihood
Estimator of p. If n = 20 and x = 3, what is the estimate? Is
this estimator unbiased?
Ans This is a Binomial model. Therefore, p.d.f. is
P [X = x] =
n
x
p x (1 − p) n−x
Because we have just a single observation, the jt. p.d.f itself is this,
and probability in this case is not 0, we can take log,
n
ln P [X = x] = ln[ ]x ln p + (n − x) ln(1 − p)
x
Maximizing w.r.t. p,
∂ ln P [X = x]
∂p
= x
p
n − x
+ (−1) = 0
1 − p
(n − x)p = x(1 − p) ⇒ np = x ⇒ ˆp = x
n
To show unbiasedness, use the fact that X is binomial,
E[X/n] = (1/n)E[X] = np/n = p
The following page contains a table of cumulative standard normal probabilities
x
Φ(x) = P [Z < x] =
For x < 0, Φ(−x) = 1 − Φ(x).
10
−∞
1 x2
− √ e 2 dx
2π