Question

Let X be the random variable for the number of heads obtained when three fair coins are tossed:

(1) What is the probability function?

(2) What is the mean?

(3) What is the variance?

(4) What is the mode?

Answer 1

1)

The random variable X = number of heads obtained when three fair coins are tossed.

The possible outcomes are,

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

The probability distribution is,

x	p(X)
0	1/8
1	3/8
2	3/8
3	1/8

2)

The expected value (mean) for the random variable x is obtained using the following formula,

From the probability distribution table,

x	p(X)	x*P(X)
0	0.125	0
1	0.375	0.375
2	0.375	0.75
3	0.125	0.375
Sum	1	1.5

3)

The standard deviation value for the random variable x is obtained using the following formula,

From the probability distribution table,

x	p(X)	x*P(X)	(x^2)*P(X)
0	0.125	0	0
1	0.375	0.375	0.375
2	0.375	0.75	1.5
3	0.125	0.375	1.125
Sum	1	1.5	3

4)

The mode is the most frequent value or most probable value in the data set. In this case,

mode = 1 and 2