Question : A random variable X has a probability density function f(x) = (x/32) : 2163580

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

52) A random variable X has a probability density function f(x) = (x/32), 0 ≤ x ≤ 8. Find a such that Pr(X ≥ a) = (1/4).

Enter your answer exactly in the reduced form b√(c), unlabeled.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Let X be a continuous random variable A ≤ X ≤ B and let f (x) be its probability density function and F (x) its cumulative distribution function. Indicate whether the following statements are true or false.

53) Pr(a ≤ X ≤ b) =

A) True

B) False

54) f(A) = 0, f(B) = 1

A) True

B) False

55) F'(x) = f(x)

A) True

B) False

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Missed work hours caused by one of a class of industrial accidents has a probability density function

f (t) = (1/8)e^-t + (3/8)e^-t/2 + (1/24)e^-t/3 where t is measured in hours.

56) What proportion of these accidents result in 5 or fewer missed work hours?

Enter just a real number to two decimal places.

57) What proportion of these accidents result in more than 9 missed work hours?

Enter just a real number to two decimal places.

58) Dr. Smith's test score distribution is characterized by the probability density function f(x) = (x(10,000 - x²)/25,000,000), 0 ≤ x ≤ 100. What percentage of people are likely to get a 60 or above on the exam? Enter just a real number to two decimal places (no units).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the problem.

59) A dart is thrown at a number line in such a way that it always lands in the interval [0,10]. Let x be the number the dart hits. Suppose the probability density function for x is given by

f(x) = (x/50), for 0 ≤ x ≤ 10.

Find P(2 ≤ x ≤ 5), the probability that it lands in [2, 5].

A) 0.03

B) 0.09

C) 0.42

D) 0.21

60) A dart is thrown at a number line in such a way that it always lands in the interval [0, 7]. Let x be the number the dart hits. Suppose the probability density function for x is given by

f(x) = (3/343)x², for 0 ≤ x ≤ 7.

Find P(2 ≤ x ≤ 5), the probability that it lands in [2, 5].

A) 0.43

B) 0.34

C) 0.03

D) 0.06

61) A random variable has probability density function f(x) = 30x²(1 - x)²(0 ≤ x ≤ 1). Compute its cumulative distribution F(x).

A) 30x² - 60x³ + 30x⁴

B) 10x³ - 15x⁴ + 6x⁵

C) 30x(1 - x)

D) 10x³ - 15x⁴ + 6x⁵ + 1

E) none of these

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

62) Suppose f(x) = k(x² + 2x) is a probability density function for a continuous random variable on the interval 0 ≤ x ≤ 3. Find the value of k and find the corresponding cumulative distribution function.

Enter just an unlabeled polynomial in x in standard form.

63) Suppose f(x) = kx^-5 is a density function for a random variable x for x ≥ 2. Find the value of k and find the corresponding cumulative distribution function.

Enter your answer exactly as a ± bx^c.

64) Given the probability density function f(x) = (1/3), determine the corresponding cumulative distribution function where 12 ≤ x ≤ 15.

Enter an unlabeled polynomial in x in standard form.

65) Given the density function f(x) = (3/64)x², 0 ≤ x ≤ 4, determine the corresponding cumulative distribution function.

Enter just an unlabeled polynomial in x in standard form.

66) Consider a square with sides of length 2 as in the diagram below. An experiment consists of choosing a point at random from the square and noting its x-coordinate. If X is the x-coordinate of the point chosen, find the cumulative distribution function of X. [Recall F(x) = Pr(0 ≤ X ≤ x).]