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Verify Property 2 of the definition of a probability density function for each of the
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- Let X be a random variable with probability density function 3 f(x) = ,a², –1 < = <1 2 Which of the following is the probability density function of Y = X – 1?arrow_forwardLet the probability density function of the random variable X is x f(x) Then Mx (t) is 0 , x = 1,2,3,... , otherwisearrow_forwardProblem 4. Suppose a line segment is divided into two pieces, the longer of length a and the shorter of length b. a + b If a then this ratio is known as the golden ratio or golden mean, denoted 1.618034. (It doesn't b. = a a +b a matter what a and b are; as long as the ratio will equal .) See the figure below: a a a+b 1 (a) Show that is a solution to the equation x2 – x – 1 = 0, and find an exact (non-decimal) expression for 4. a +b (Hint: In the equation a let b = 1 and a = x, so b' = x = 4.) a (b) Consider the sequence a, defined in the following way: 1 a2 = 1+ 1 1 1 a4 = 1+ 1+ az = 1+ 1+ a1 = 1+1, az = 1+ 1 1+ 1+1 1+1 1 1+ 1+1 1 1+ 1 1 + 1+1 This sequence has a limit (you do not need to prove this), given by 1 L = 1+ 1 1+ 1 1 + 1+ 1+.. What is this limit? (Hint: First explain why L = 1+ .) Problem 5. Show that lim Vn! = o0. (Hint: Verify that n! > (n/2)"/2 by observing that half of the factors are greater than or equal to n/2. You may want to consider the cases where n is even andn…arrow_forward
- Verify Property 2 of the definition of a probability density function over the given interval. f(x) = - 20 (3,7] What is Property 2 of the definition of a probability density function? O A. The area under the graph of f over the interval [a,b] is 1. O B. The area under the graph of f over the interval [a,b] is b. OC. The area under the graph of f over the interval [a,b] is a. Identify the formula for calculating the area under the graph of the function y = f(x) over the interval [a,b]. Choose the correct answer below. O A. b О В. а Sx) dx = [F(x); = F(b) –- F(a) J(x) dx = [F(x)!% = F(a) – F(b) a b. OC. a O D. b f(x) dx = [F(x)] = F(b) – F(a) (x) dx = = F(a) - F(b) Substitute a, b, and f(x) into the left side of the formula from the previous step. 1 area = 20x dxarrow_forwardIf a probability density function for X is f(x) define F(x) over the whole real line. = 2(x+1)² for 0 < x < 2, find F(x). Make sure youarrow_forwardLet X be a random variable with probability density function [ a + bx², 0arrow_forwarda) What must c be for the function to be a probability density function? b) Calculate F(1).arrow_forwardThe function f(x) is the probability density function, that describes the random variable travel time X. f(x) = {xe®, x>0 0, X ≤0 a.) What is the probability that the travel time is less than 2.5 hours? b.) What is the probability that the travel time is between 1 to 2.5 hours? c.) Find the mean and variance of the given probability density functionarrow_forwardLet X be a random variable with the probability density function. f(r) = 2c for x = 1,2,3, 4, ..., 0 for some constant c. What is the value of c What is the probability that X is even?arrow_forward1. What two properties must a function f that is a probability density function exhibit?arrow_forwardlet X denotes the percentage of time out of a 40-hour workweek that a call center agent is directly serving a client by answering phone calls. Suppose that X has a probability density function defined by f(x) =3x² for 0 ≤ x ≤ 1. Find the mean and variance of X. Interpret the results.arrow_forwardIf X be a random variable with probability density function f) = 2x Osxs1 V(1/2) 1/2 1 2/3arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill