Exercises. 1. Each of the following tables lists certain values of X and their probabilities

1. Each of the following tables lists certain values of X and their probabilities. Determine if each of them satisfies the two conditions required for a valid probability distribution.

X	P (x)
	0.16 0.00 0.43 0.41

a) b) c)

X	P (x)
	-0.39 0.67 0.31 0.28

X	P (x)
	0.22 0.23 0.65

2. For each case, list the values of x and P (x) and examine if the specification represents a probability distribution. If does not, state what properties are violated

a) for

b) for

c) for

d) for

3. The following table gives the probability distribution of a discrete random variable X.

X	0 1 2 3 4 5
P (x)	0.03 0.13 0.22 0.31 0.19 0.12

a) Draw the probability function.

b) Calculate and draw the cumulative probability function.

c) Find ; ; ;

d) Find the probability that x assumes a value less than 3

e) Find the probability that x assumes a value in the interval 2 to 4.

4. Despite all safety measures, accidents do happen at the factory. Let X denote the number of accidents that occur during a month at this factory. The following table lists the probability distribution

of X.

X	0 1 2 3 4
P (x)	0.25 0.30 0.20 0.15 0.10

a) Draw the probability function.

b) Calculate and draw the cumulative probability function.

c) Determine the probability that the number of accidents that will occur during a given month at this company is exactly 4.

d) What is the probability that number of accidents will be at least 2?

e) What is the probability that number of accidents will be less than 3?

f) What is the probability that number of accidents will be

between 2 to 4?

g) Two month are chosen at random. What is the probability that on both

of these months there will be fewer than two accidents?

5. Let X be the number of shopping trips made by family during a month. The following table lists the frequency distribution of X of 1000 families.

X	4 5 6 7 8 9 10
f	70 180 240 210 170 90 40

a) Draw the probability function.

b) Calculate and draw the cumulative probability function.

c) Find the following probabilities ; ; ; .

6. The following table lists the probability distribution of the number of phone calls received per 10-minute period at an office.

Number of phone calls	0 1 2 3 4
Probability	0.12 0.26 0.34 0.18 0.10

Let X denote the number of phone calls received during a certain 10-minute period at this office.

a) Find the probabilities: ; ; ; .

b) Two 10-minute period are chosen at random. What is the probability that at least one of them there will be at least one received call?

7. In successive rolls of a fair die, let X be the number of rolls until the

first 6. Determine the probability function.

8. In a tennis championship, player A competes against player B in consecutive sets and the game continues until one player wins three sets. Assume that, for each set P (A wins) =0.4, P (B wins) =0.6, and the outcomes of different sets are independent. Let X stand for the number of sets played.

a) List the possible values of X and identify the basic outcomes associated with each value.

b) Obtain the probability distribution of X.

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