Question

State whether the following statements are true or false. Give a short proof or a counter  example in support of your answers:

(a) Poisson distribution is a limiting case of binomial distribution for n→∞, p→1 and np→∞.

(b) For two independent events A and B, if P(A) = 2.0 and P(B) = ,4.0 then (A∩ B) = .6.0

(c) If H₀: P ≤ 6.0 and X ~ B(n, p) n -known and p unknown and H₁ :µ = µ₀ where 

X ~ N (µ,σ²)σ² unknown, then H₀

and H₁ are simple null hypothesis. 

(d) Frequency density of a class for any distribution is the ration of total frequency to  class width. 

(e) If X and Y are independent r.v.s with M_x(t) and M_Y(t) as their m.gf's respectively, then M_X+Y (t) = M_X (t) M_Y(t).

Answer 1

a) true

 If n→∞, p→1 and np→∞ for binomial distribution:

$P(x=k)=\frac{n!}{k!(n-k)!}p^k(1-p)^{n-k}$

This can be rewritten as

$\frac{\mu^k}{k!}\frac{n!}{(n-k)!n^k}(1-\frac{\mu}{n})^n(1-\frac{\mu}{n})^{-k}$

$lim(n\to \infin)(\frac{\mu^k}{k!}\frac{n!}{(n-k)!n^k}(1-\frac{\mu}{n})^n(1-\frac{\mu}{n})^{-k})=\frac{\mu^k}{k!}\cdot1\cdot e^{-\mu}\cdot1=\frac{\mu^ke^{-\mu}}{k!}$

b) false

For this two independent events:

P(A∩ B)=$P(A)\cdot P(B)=0.2\cdot0.4=0.08$

c) false

Simple hypotheses are ones which give probabilities to potential observations.

So, only H₀ is simple null hypothesis.

d) false

For a set of grouped data, the frequency density of a class is defined by the ration of frequency in the class to class width.

e) true

By Proposition:

Suppose X1, ..., Xn are n independent random variables, and the random variable Y is defined by

Y = X1 + ... + Xn. Then m_{Y (t)} = m_{X1 (t)} · ... · m_{Xn (t)}.

So, if Z=X+Y then:

M_Z =M_X+Y = M_X · ... · M_Y