Question

Each of n people (whom we label 1, 2, . . . , n) are randomly and independently assigned a number from the set {1, 2, 3, . . . , 365} according to the uniform distribution. We will call this number their birthday. (a) Describe a sample space Ω for this scenario. Let j and k be distinct labels (between 1 and n) and let Ajk denote the event that the corresponding people share a birthday. Let Xjk denote the indicator random variable associated to Ajk. (b) Write A12 as a subset of Ω. (c) Tabulate the joint PMF for X12 and X13. Compute the PMF for the product X12X13. (d) Tabulate the joint PMF for X12 and X34. Compute the PMF for the product X12X34. (e) Are A12 and A34 independent? Are they independent conditioned on A13? (f) Are A12 and A13 independent? Are they independent conditioned on A23? (g) Compute the expected number of pairs of people who share a birthday (hint: write this the number as a sum of Xjks). (h) Compute the second moment and variance of the number of pairs of people who share a birthday

Answer 1

Given each of people (whom we label ) are randomly and independently assigned a number from the set according to the uniform distribution.

a) The sample space is

Also, the cardinality is .

b) Persons can share the birthday in ways.

c) We have . So we have

Similarly,

Now the joint PMF is

We can see that .

From the table we can see that . So the PMF of the product,

d) Now the joint PMF of is

We can see that .  The PMF of the product,

e) We can see that .

Hence are independnet.

We are required to solve only 4 parts. Please post the remaining questions as another post.
We do not get any additional amount for solving more.