Question

Which of the following functions are probability mass functions? For those that are not, find (if possible) a constant a so that a · p(ω) is a probability mass function.

a. p(ω) = ω 2 55 , ω = 1, 2, 3, 4, 5

b. p(ω) = 1 3 2 3 ω , ω = 3, 4, 5, 6, . . .

c. p(ω) = 1 for each ω in a nine-member set Ω.

d. p(ω) = 1 for each ω in a countably infinite set Ω.

e. p(ω) = ω, ω = 1, 2, 3, 4, . . . , N

f. p(ω) = 1 4 ω , ω = 0, 1, 2, 3, 4, . . .

g. p(ω) = 1 ω , ω = 1, 2, 3, 4, . . .

h. p(ω) = 1 3 (ω − 2), ω = 0, 1, 2, 3, 4

Answer 1

Answer:

a) p(w) = w^2/55 = ( 1+4+9+16+25)/55 = 55/55 = 1   Yes it is PMF

b) p(w) = (1/3)(2/3)^w = (1/3)(2/3)^3+(1/3)(2/3)^4+(1/3)(2/3)^5+(1/3)(2/3)^6+........Its a GP

= (1/3)(2/3)^3 / (1 -2/3)

= 1/3 * 8/27 * 3

=8/27   : Not a PMF , a = 27/8

c) p(w) =1 Not a pmf , a = 1/9

Explanation: It is actually for nine members : (1+1+1+1+1+1+1+1+1) = 9

Hence a =1/9 is chosen to make it as 1.

d) p(w) =1 Not a pmf , a = 1/N

Explanation: It is actually for countably infinite , so let N be the count : (1+1+1+1+1+1+1+1+1+........+1)----> N times = N

e) p(w) = w

= 1+2+3+4,.......N

= N+1

f) p(w) = (1/4)^w

= from b we can conclude f is also not a PMF

g) p(w) = 1/w

= 1/1+1/2+1/3 + 1/4....

= not a PMF

h) p(w) = 1/3(w-2)

= 1/3(0-2)+ 1/3(1-2)+ 1/3(2-2)+ 1/3(3-2)+ 1/3(4-2)

= -2/3+-1/3+0+2/3

=1/3