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Verify Property 2 of the definition of a probability density function for each of the
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- If 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_forwardVerify 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_forwarda) What must c be for the function to be a probability density function? b) Calculate F(1).arrow_forward
- The 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_forwardIf X be a random variable with probability density function f) = 2x Osxs1 V(1/2) 1/2 1 2/3arrow_forward1. What two properties must a function f that is a probability density function exhibit?arrow_forward
- A random variable X has probability density function (x² + 2x + 4) for 0 < x < 3 f(x) = otherwise. Which of the following gives the cumulative distribution function F(x)?arrow_forwardLet X1, X2, and X3 be independent random variables from (-1, 1). Find the probability density function and the expected value of the random variable [X(1) + X(2)]/2.arrow_forwardWhich of the following tables shows a valid probability density function? Select all correct answers. Select all that apply: x P(X=x) 0 14 1 38 2 38 x P(X=x) 0 18 1 18 2 34 x P(X=x) 0 −310 1 110 2 15 3 1 x P(X=x) 0 0.74 1 0.0 2 0.54 x P(X=x) 0 110 1 15 2 35 x P(X=x) 0 0.18 1 0.76 2 0.0arrow_forward
- Suppose that the random variable X has the probability density function f(x) = { c(1- x2) for - 1< x 1 elsewhere Find the probability that X is in the interval of [0,1/2]arrow_forwardLet X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.arrow_forwardSuppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.arrow_forward
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill