Question

Suppose that we go sequentially through a group of people asking each in turn to pick one of 5 options (e.g., "rock," "paper," "scissors," "fire," or "water") at random. Let X_i (for 0 ? i ? 4) be the random variable giving the number of additional people we have to ask before getting an option that has not been picked yet. (So X₀ is the number of people we have to ask to get the first option, which will be 1 person; X₁ the number of additional people we ask to get the second, etc.)

Find the following:

P {X_i = k}
E [Xi]

Additionally, if X is the number of people we will have to ask before we've heard all all of the options, what is E[X]?

Answer 1

X0 is the number of people we have to ask to get the first option, which will be 1 person.

So, P{X0 = 1} = 1

X1 the number of additional people we ask to get the second option with p = 4/5 (there are 4 options remaining out of 5 options). By Geometric distribution,

P{X1 = k} = (1/5)^k-1 (4/5)

X2 the number of additional people we ask to get the third option with p = 3/5 (there are 3 options remaining out of 5 options). By Geometric distribution,

P{X2 = k} = (2/5)^k-1 (3/5)

X3 the number of additional people we ask to get the fourth option with p = 2/5 (there are 2 options remaining out of 5 options). By Geometric distribution,

P{X3 = k} = (3/5)^k-1 (2/5)

X4 the number of additional people we ask to get the fifth option with p = 1/5 (there are 1 option remaining out of 5 options). By Geometric distribution,

P{X4 = k} = (4/5)^k-1 (1/5)

Thus,

P{X0 = 1} = 1

P{X1 = k} = (1/5)^k-1 (4/5)

P{X2 = k} = (2/5)^k-1 (3/5)

P{X3 = k} = (3/5)^k-1 (2/5)

P{X4 = k} = (4/5)^k-1 (1/5)

By Geometric distribution, E[Xi] = 1/p_i

Thus,

E[X0] = 1

E[X1] = 1/(4/5) = 5/4

E[X2] = 1/(3/5) = 5/3

E[X3] = 1/(2/5) = 5/2

E[X4] = 1/(1/5) = 5/1 = 5

X is the number of people we will have to ask before we've heard all all of the options, then

X = X0 + X1 + X2 + X3 + X4

E[X] = E[X0 + X1 + X2 + X3 + X4]

= E[X0] + E[X1] + E[X2] + E[X3] + E[X4]

= 1 + (5/4) + (5/3) + (5/2) + 5

= 137/12

= 11.4167