Question

					x	P(X=x)
					0	0.44
The incomplete table at right is a discrete random variable x's probability distribution, where x is the number of days in a week people exercise.  Answer the following:					1	0.18
					2	0.02
					3
(a)	Determine the value that is missing in the table.				4	0.03
					5	0.13
					6	0.07
(b)	Explain the meaning of " P(x < 2) " as it applies to the context of this problem.				7	0.04
(c)	Determine the value of P(x > 4):
(d)	Find the probability that x is at least 1.
(e)	Find the mean (expected value) and standard deviation of this probability distribution.

Answer 1

(a) Here sum of all the probabilites shall be equal to 1.

= 1

0.44 + 0.18 + 0.02 + P(3) + 0.03 + 0.13 + 0.07 + 0.04 = 1

P(3) = 1 - 0.91 = 0.09

(b) Here the meaning of P(x < 2) is values of x that is less than 2. here it means the probability that number of days people exercise less than 2 in a week.

(c) P(x < 4) = P(0) + P(1) + P(2) + P(3) = 0.44 + 0.18 + 0.02 + 0.09 = 0.73

(d) P(x at least 1) = 1 - P(x is less than 1) =1 - P(0) = 1 - 0.44 = 0.56

(e) Mean = = 0 * 0.44 + 1 * 0.18 + 2 * 0.02 + 3 * 0.09 + 4 * 0.03 + 5 * 0.13 + 6 * 0.07 + 7 * 0.04 = 1.96

Standard deviation = Sqrt (Variance)

Variance = - ()²

= 0 * 0.44 + 1 * 1 * 0.18 + 2 * 2 * 0.02 + 3 * 3 * 0.09 + 4 * 4 * 0.03 + 5 * 5 * 0.13 + 6 * 6 * 0.07 + 7 * 7 * 0.04 = 9.28

Variance = 9.28 - 1.96 * 1.96 = 5.4384

Standard deviation = sqrt(5.4384) = 2.332