Question

Suppose you have an experiment where you flip a coin three times. You then count the number of heads.

1. State the random variable.
2. Write the probability distribution for the number of heads.
3. Draw a histogram for the number of heads.
4. Find the mean number of heads.
5. Find the variance for the number of heads.
6. Find the standard deviation for the number of heads.
7. Find the probability of having two or more number of heads.
8. Is it unusual to flip two heads?

Answer 1

A coin tossed three times.

The sample space is given by

S = { (H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, H, H), (T, H, T), (T, T, H), (T, T, T)}

The random variable X is defined as

X: number of heads in three tosses.

X takes values 0,1, ,2 and 3.

The probability distribution of X is

X	0	1	2	3	Total
P(X=x)	1/8	3/8	3/8	1/8	1

E(X) = 0 * 1/8 + 1 /3/8 + 2*3/8 + 3 *1/8

= 3/8 + 6/8 + 3/8 = 1.5

Mean= 1.5.

Variance of number of heads is given by

E(X²) = 0 * 1/8 + 1* 3/8 + 4* 3/8 + 9* 1/8

= 3/8 + 12/8 + 9/8

= 3

Var(X) = 3 - 2.25 = 0.75

Var(X) = 0.75

Standard deviation of number of heads

SD(X) = sqrt(Var(X)) = sqrt(0.75) = 0.8660.

P ( Having two or more number of heads) = P ( X > = 2)

= P(X=2) + P(X=3)

= 3/8 + 1/8

=0.5.

P ( Having two number of heads ) = P(X=2) = 3/8 = 0.375.

Since P ( X=2) = 0.375 > 0.05 hence it is not unusual.

For drawing histogram we need to draw random sample from binomial distribution with n = 3 and p = 0.5

by using R

> r= rbinom(100,3,0.5)
> hist(r)
> table(r)
r
0 1 2 3
14 30 41 15