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Show that the probability distribution of x satisfies the properties of a discrete probability

Show that the probability distribution of x satisfies the properties of a discrete probability Exercise 5.12 METHODS AND APPLICATIONSSuppose that the probability distribution of a random variable x can be described by the formula

P(x) = x________________________________________15

for each of the values x = 1, 2, 3, 4, and 5. For example, then, P(x = 2) = p(2) =2/15.

(a) Write out the probability distribution of x. (Write all fractions in reduced form.)

x 1 2 3 4 5

P(x) ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________

________________________________________

(b) Show that the probability distribution of x satisfies the properties of a discrete probability distribution.(Round other answers to the nearest whole number. Leave no cells blank – be certain to enter “0” wherever required.)

P(x) ≥ for each value of x.

(c) Calculate the mean of x. (Round your answer to 3 decimal places.)

µx

(d) Calculate the variance, σ2x , and the standard deviation, σx. (Round your answer σx2 in to 3 decimal places and round answer σx in to 4 decimal places.)

σx2

σx

2.award:3.43 out of5.00 points

Exercise 5.23 METHODS AND APPLICATIONSSuppose that x is a binomial random variable with n = 5, p = 0.3, and q = 0.7.

(b) For each value of x, calculate p(x), and graph the binomial distribution. (Round final answers to 5 decimal places.)

p(0) = , p(1) = , p(2) = , p (3) = ,p(4) = , p(5) =

(c) Find P(x = 3). (Round final answer to 5 decimal places.)

P(x=3)

(d) Find P(x ≤ 3). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x ≤ 3)

(e) Find P(x < 3). (Do not round intermediate calculations. Round final answer to 5 decimal places.) P(x < 3) = P(x ≤ 2) (f) Find P(x ≥ 4). (Do not round intermediate calculations. Round final answer to 5 decimal places.) P(x ≥ 4) (g) Find P(x > 2). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x > 2)

(h) Use the probabilities you computed in part b to calculate the mean, μx, the variance, σ 2x, and the standard deviation, σx, of this binomial distribution. Show that the formulas for μx , σ 2x, and σx given in this section give the same results. (Do not round intermediate calculations. Round final answers to µx and σ 2x in to 2 decimal places, and σx in to 6 decimal places.)

µx

σ2x

σx

(i) Calculate the interval [μx ± 2σx]. Use the probabilities of part b to find the probability that x will be in this interval. Hint: When calculating probability, round up the lower interval to next whole number and round down the upper interval to previous whole number. (Round your answers to 5 decimal places. A negative sign should be used instead of parentheses.)

The interval is [ , ].

P( ≤ x ≤ ) =

3.award:3 out of3.00 points

MC Qu. 14 The mean of the binomial distribution is equ…The mean of the binomial distribution is equal to:

p

np

(n) (p) (1-p)

px (1-p)n-x5.award:3 out of3.00 points

MC Qu. 25 A fair die is rolled 10 times. What is the p…A fair die is rolled 10 times. What is the probability that an odd number (1, 3, or 5) will occur less than 3 times?

.1550

.8450

.0547

.7752

.11728.award:3 out of3.00 points

MC Qu. 31 If n = 20 and p = .4, then the mean of the b…If n = 20 and p = .4, then the mean of the binomial distribution is

.4

4.8

8

1210.award:3 out of3.00 points

MC Qu. 36 The probability that a given computer chip w…The probability that a given computer chip will fail is 0.02. Find the probability that of 5 delivered chips, exactly 2 will fail.

.9039

.0000

.0922

.003812.award:3 out of3.00 points

MC Qu. 38 In the most recent election, 19% of all elig…In the most recent election, 19% of all eligible college students voted. If a random sample of 20 students were surveyed:Find the probability that exactly half voted in the election.

.4997

.0014

.0148

.000013.award:3 out of3.00 points

MC Qu. 39 In the most recent election, 19% of all elig…In the most recent election, 19% of all eligible college students voted. If a random sample of 20 students were surveyed:Find the probability that none of the students voted.

.0148

.4997

.0014

.000021.award:3 out of3.00 points

MC Qu. 55 For a random variable X, the mean value of t…For a random variable X, the mean value of the squared deviations of its values from their expected value is called its

Standard Deviation

Mean

Probability

Variance25.award:3 out of3.00 points

MC Qu. 62 If the probability distribution of X is:&nbs…If the probability distribution of X is:

What is the expected value of X?

2.25

2.24

1.0

5.0

26.award:0 out of3.00 points

MC Qu. 63 If the probability distribution of X is:&nbs…If the probability distribution of X is:

What is the variance of X?

5.0→1.0

2.25

2.24

28.award:0 out of3.00 points

MC Qu. 66 A vaccine is 95 percent effective. What is t…A vaccine is 95 percent effective. What is the probability that it is not effective for, more than one out of 20 individuals?

.3774

.7359→.2641

.3585P(X ≥ 2) = 1 – [P(X = 0) + p(X = 1)]P(X ≥ 2) = 1 – [(.3585) + (.3774)] = .2641

29.award:3 out of3.00 points

MC Qu. 67 If the probability of a success on a single …If the probability of a success on a single trial is .2, what is the probability of obtaining 3 successes in 10 trials if the number of successes is binomial?

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