4. (i) Let a be a positive constant and f(x) = ax² e −4x x = R. Find a such that f(x) is a probability density function. [6 Marks] (ii) Let X be a random variable with probability density function in (i) (a) Find (A), the characteristic function of the random variable X. (b) Using (A), calculate E(X) and Var(X). [15 Marks] [14 Marks]
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![4. (i) Let a be a positive constant and
f(x) = ax² e −4x
x = R.
Find a such that f(x) is a probability density function.
[6 Marks]
(ii) Let X be a random variable with probability density function in (i)
(a) Find (A), the characteristic function of the random variable X.
(b) Using (A), calculate E(X) and Var(X).
[15 Marks]
[14 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c7573-6777-48ff-9bfb-9b3df1a769a6%2Fcbebaf3a-5486-4c82-b055-dd57dec47cd0%2Fijyfwon_processed.jpeg&w=3840&q=75)
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- Let X be a random variable with discrete pdf f(x) = x/8 if x = 1, 2, 5, and zero otherwise. Find: (a) E(X).The probability density functionLet X and Y have the joint probability density functionf(x, y) = 5, for 0 < x < y < 1.(i) Find the marginal probability density function of X and Y.(ii) Find the conditional probability density function of X, given Y = y.(iii) Find the conditional mean of X, given Y = y.(iv) Find the conditional variance of X, given Y = y.
- Let f(x) = k(4x –x²) if 03. (a) For what value of k is f(x) a probability density function? (b) For that value of k, find P (x>2). (c) Find the mean.The probability mass function for a discrete random variable X is defined as ((1+0)-(x) 0; x = 0,1,2,3,..., η (x) = {(1+0) (2) *; fx(x) 0; e.w. where 0 > 0. Show that it is probability mass function. Find its mean and variance.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…
- Suppose a number is chosen randomly from the interval [0, 4]. Let X be the value chosen. (So X is a uniform random variable over the interval [0, 4].) (a) What is the probability density function of X? (b) Find E(eX + X³).Q2 Let (X₁, X₂) be jointly continuous with joint probability density function = { -(x₁+x2), 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). Q2(iv.) Let Mx, (t) = Mx₂ (t) = (¹, for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. 1 - Q2(v.) Let Y₂ = X₁ – X2, and Mx, (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…1. A continuous random variable X has a probability density function (PDF) p(x) = k( 8-x^2/2) on the interval [0, 4] (a) Find k such that p(x) is a valid PDF. (b) Find P(X ≤ 1). (c) Find the mean, µ, of X. (d) Find the variance, σ 2 , of X.
- Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-∞,0) is a function f such that f(2) 20 and Lsla) = 1. (a) Determine which of the following functions are probability density functions on the (-00, 00). (1-1 00 (b) We can also use probability density functions to find the expected value of the outcomes of the event - if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. zf(z) dz yields the expected value for a density f(x) with domain on the real mumbers.) Find the expected value for one of the valid probability densities above.(b) Let X and Y have joint probability density function fxx (2,3)=ô 0,2+y<1 0, otherwise. (i) Show that k = 2. (ii) Find the marginal probability density functions of X and Y. (iii) Find the conditional probability density functions fxx (xly) and fy|x (ylx). (iv) Find the conditional expectation E(XY) and the conditional variance var (XY). (v) Find E (E (XY)). Hence verify the formula E (X) = E (E (X|Y)). (vi) Are X and Y independent? Give reasons.Example of (a) , (b) is (12),(24),(1),(6) Example of (c) (i) or (ii) Thanks
