Question

Solve the following. 16 - 20. Find the mean of the probability distribution of the random variable x, which can take only the values 1, 2, 3 given that P(1)= 0.3 , P(2)= 0.5, and P(3)= 0.2​

Answer 1

Answer:

fair coin is tossed twice. Let X be the number of heads that are observed.

Construct the probability distribution of X.

Find the probability that at least one head is observed.

Solution:

The possible values that X can take are 0, 1, and 2. Each of these numbers corresponds to an event in the sample space S={hh,ht,th,tt} of equally likely outcomes for this experiment: X = 0 to {tt}, X = 1 to {ht,th}, and X = 2 to {hh}. The probability of each of these events, hence of the corresponding value of X, can be found simply by counting, to give

xP(x)00.2510.5020.25

This table is the probability distribution of X.

“At least one head” is the event X ≥ 1, which is the union of the mutually exclusive events X = 1 and X = 2. Thus

P(X≥1)=P(1)+P(2)=0.50+0.25=0.75

A histogram that graphically illustrates the probability distribution is given in Figure 4.1 "Probability Distribution for Tossing a Fair Coin Twice".