Question

4 fair coins are tossed. Let X be the number of heads and Y be the number of tails. Find Var(X-Y)

Solution: 3.5

Why?

Answer 1

Z = X - Y

can take values - 4, - 2,, 0, 2. ,4

For Z = - 4:

X = 0, Y = 4. P(X=0,Y=4)= Probability of getting 4 Tails = 1/2⁴= 0.0625

For Z = - 2:

X = 1, Y = 3. P(X=1,y=3)= Probability of getting 1 Head,3 Tails = 4 X 1/2⁴= 4 X 0.0625= 0.25

For Z = 0:

X = 2, Y = 2. P(X=2,Y=2)= Probability of getting 2 Head, 2 Tails = 6 X 1/2⁴= 6 X 0.0625= 0.3750

For Z = 2:

X = 3, Y = 1. P(X=3,Y=)= Probability of getting 3 Heads, 1 Tail = 4 X 1/2⁴= 4 X 0.0625= 0.25

For Z = 4:

X = 4, Y = 0. P(X=4,Y=0)= Probability of getting 4 Heads, 0 Tail = 1/2⁴= 0.0625

So, we get:

z      p                zp                z² p

- 4   0.0625     - 0.25                 1

- 2    0.25        - 0.5                    1

0 0.3750       0                       0

2 0.25        0.5                      1

4    0.0625     0.25                     1

--------------------------------------------------

Total                0                     4

So,

E(Z) = 0

E(Z²) = 4

Var(Z) = E(Z²) - (E(Z))²

      = 4