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Question : Find the value of k that makes f(x) = kx^2 a probability density function on the interval : 2163577

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

35) Find the value of k that makes f(x) = kx^{2} a probability density function on the interval 0 ≤ x ≤ 1.

Enter just an integer.

36) Find the value of k that makes f(x) = 3e^{-kx} a probability density function on the interval x ≥ 0.

Enter just an integer.

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

37) A random variable X has a cumulative distribution function F(x) = 1 - (1/x^{2}) (x ≥ 1). Find Pr(a ≤ X ≤ 5).

A) 1 - (1/a^{2})

B) (1/a^{2})

C) (1/a^{2}) - (1/25)

D) (24/25) - (1/a^{2})

E) none of these

38) Let X be a continuous random variable with a cumulative distribution function F(x) = 1 - e^{-x^2} (x ≥ 0). Find Pr(1 ≤ X ≤ 2).

A) e^{-1} - e^{-2}

B) e^{-1} - e^{-4}

C) e^{-1} - e^{-2} - 2

D) 1 - e^{-1} - e^{-2}

E) none of these

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

39) The probability density function for a random variable X is f(x) = 3x^{-4}, x ≥ 1. Find Pr(2 ≤ X).

Enter just a reduced fraction.

40) The probability density function for a random variable X is f(x) = (3/4)(2x - x^{2}), 0 ≤ x ≤ 2 . Find Pr(0 ≤ X ≤ 1).

Enter just a reduced fraction.

41) The probability density function for a random variable X is f(x) = (2lnx/(ln4)^{2} x) , 1 ≤ x ≤ 4. Find Pr(1 ≤ X ≤ 2).

Enter just a reduced fraction.

42) The probability density function of a continuous random variable X is f(x) = (3/2)x - (3/4)x^{2} , 0 ≤ x ≤ 2. Is this the graph of f(x) with the shaded area corresponding to Pr((1/2) ≤ x ≤ (3/2))?

Enter "yes" or "no".

43) If f(x) = (1/8)x is a probability density function for 0 ≤ x ≤ 4, find F(x), the corresponding cumulative distribution function and use it to find Pr(1 ≤ X ≤ 3).

Enter just a reduced fraction representing Pr(1 ≤ X ≤ 3). Do not label.

44) If f(x) = 6x(1 - x) is a probability density function for 0 ≤ x ≤ 1, find F(x), the corresponding cumulative distribution function and use it to find Pr((1/2) ≤ X ≤ 1).

Enter just a reduced fraction representing Pr((1/2) ≤ X ≤ 1). Do not label.

45) A random variable X has a cumulative distribution function F(x) = 1 - (1/(x + 1)^{2}) for x ≥ 0. Find Pr(1 ≤ X ≤ 4).

Enter just a reduced fraction.

46) Suppose f(x) = (1/x^{2}) is a probability density function for x ≥ 1. Find Pr(2 ≤ X ≤ 10).

Enter just a reduced fraction.

47) Determine the probability of an outcome of the probability density function f(x) = 4x^{3} being between (1/4) and (1/2) where 0 ≤ x ≤ 1.

Enter just a reduced fraction.

48) Determine the probability of an outcome of the probability density function f(x) =12x^{2} - 12x^{3} being between (1/2) and 1 where 0 ≤ x ≤ 1.

Enter just a reduced fraction.

49) A random variable X has a density function f(x) = (1/ln16)⋅(1/x) , 1 ≤ x ≤ 16. Find a such that Pr(1 ≤ X ≤ a) = (3/4).

Enter just an integer, no labels.

50) A random variable X has a density function f(x) = (24/x^{3}), 3 ≤ x ≤ 6. Find b such that Pr(X ≤ b) = 0.4.

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

51) A random variable X has a cumulative distribution function F(x) = (x/5) - 2, 10 ≤ x ≤ 15. Find a such that Pr(a ≤ X ≤ 15) = (2/3).

Enter just a reduced fraction of form (a/b).