{ ω 11 , ω 12 , ω 13 ,... ω 1 k } is

{ ω , ω , ω ,... ω k } is called inverse image of x . That is X ( ) ω { ω , ω , ω ,... ω k } is an event in S Illustration .

Suppose a coin is tossed once. The sample space consists of two sample points H (head) and T (tail). That is S T H Let X S : →  be the number of heads Then X T , and X H . Thus X is a random variable that takes on the values and .

If X ( ) ω denotes the number of heads, then X ( ) for =Tail for = Head Example . Suppose two coins are tossed once. If X denotes the number of tails, (i) write down the sample space (ii) find the inverse image of (iii) the values of the random variable and number of elements in its inverse images. (i) The sample space S H T H T i ω ω i X ω X ω S X Sample space Real numbers line  ω ω ω ω ,  k Fig.

. Probability Distributions That is S TT TH HT HH (ii) Let X S : →  be the number of tails Then X TT = ( Tails) X TH = ( Tail) X HT = ( Tail) and X HH = ( Tails). Then X is a random variable that takes on the values , and . Let X ( ) ω denote the number of tails, this gives X TT HT TH HH if if if The inverse images of is TH HT .