10 pts Numbers are drawn from the set of integers 1 2 3

(10 pts) Numbers are drawn from the set of integers {1, 2, 3, . . . , 21} uniformly at random, without replacement. Let X be the number of draws among the first ten draws that are divisible by seven.

(a) (5 pts) What is E(X)?

(b) (5 pts) What is Var(X)?

Solution

numbers are drawn without replacement is equivalent to 10 numbers drawn at random at once

Thus,

P(X = 0) = all numbers selected from the non-multiples of 7

= ¹⁸C₁₀ / ²¹C₁₀

= 33/266

P(X = 1)

= ¹⁸C₉ * ³C₁ / ²¹C₁₀

= 55/133

P(X = 2)

= ¹⁸C₈ * ³C₂ / ²¹C₁₀

= 99/266

P(X = 3)

= ¹⁸C₇ * ³C₃ / ²¹C₁₀

= 12/133

Thus, expected value = 0(33/266) + 1(55/133) + 2(99/266) + 3(12/133)

= 10/7

E[X] = 10/7

Now,

Var[X] = (33/266)*[0 - 10/7]² + (55/133)*[1 - 10/7]² + (99/266)*[2 - 10/7]² + (12/133)*[3 - 10/7]²

= 33/49

Hope this helps.