f(x)=3x² 2 0 ≤ x ≤ 1 a) Verify that (x) is a probability density function b) Find the CDF F(x) c) Find Pr (0.25 ≤ X ≤ 0.75) AQAAAAAAAAAAAAAAAAAAAAA
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- Help me pleasea) Find k such that the function is a probability density function over the given interval. b) Then write the probability density function. ƒ(x) = [5, 20] k -, Xlet x be a continuous random Variable with probability density fimction 0Q1) CDF, PDF, Expectation and Variance The cumulative distribution function (CDF) of random variable Y is y 1. a) Find the probability density function, fy(y), of random variable Y. b) Show that f fy (y)dy = 1. c) What is E[Y]? d) What is VAR[Y]?Sketch the graph of the probability density function over the indicated interval. F(x) = 1/20, [0, 20] Find the indicated probabilities. a) P(0 < x < 12) b) P(8 < x < 12) c) P(13 < x < 20) d) P(x ≥ 5)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 youlet x be a continuous random variable with probability density function Find Sb – 2x, 0Let x be a continuous random variable that can take any value in the interval [0,3]. Let its probability density function be f(x) = {cx² 0a) What must c be for the function to be a probability density function? b) Calculate F(1).Let X be a continuous random variable with probability density function 1 f (x) : T(1+ x2) Define Y 4. What is the mean of y?Suppose a random variable X has a probability density function as shown in img1.jpg. a) Find the value of k that makes this a probability density function b) Find P(0.4 ≤ X ≤ 1). c) Find P(X ≤ 0.4 | X ≤ 0.8) d) Find F(b) = P(X ≤ b) and sketch the graph of this function.Q2 Let (X1, X₂) be jointly continuous with joint probability density function e-(x1+x2), 0 f(x1, x₂) = x₁ > 0, x₂ > 0 otherwise. Q2 (i.) Sketch(Shade) the support of (X₁, X₂). Q2 (ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2 (iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and then find the probability density function of Y₁, i.e., fy, (y). 1 = Q2 (iv.) Let Mx, (t) = Mx₂ (t) (1 t), for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. Q2(v.) Let Y₂ = X₁ — X₂, and Mx₁ (t) = Mx₂(t) = (1 t). Find the moment generating function of Y2, and using the moment generating function of Y₂, find E[Y₂]. Q2 (vi.) Using the bivariate transformation method, find the joint…SEE MORE QUESTIONSRecommended textbooks for youCalculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage Learning
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