a) Suppose that, for -1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {11₁ - α(1 — 2e-x¹) (1 - 2e¯*²)]e*1-*² 0, ,0 ≤ x₁,0 ≤ x₂. otherwise i) Find the marginal distribution of X₁. ii) Find E(X₁X₂).
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![QUESTION 3
a) Suppose that, for -1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by
f(x₁, x₂) = {11 - α(1 - 2e¯¹) (1 - 2e-*²)]e*¹-*₂
,0 ≤ x₁,0 ≤ x₂.
otherwise
i) Find the marginal distribution of X₁.
ii) Find E(X₁X₂).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc713c345-a6c2-46f2-a3e8-826737fb1de0%2Fe39724e1-a14d-45a1-b948-95dee296b6bb%2F7kpnd7e_processed.jpeg&w=3840&q=75)
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- 1. Let f be the probability density function of a continuous random variable X whose range is (-0, 00), where if a sxsb f(x) ={32 0 otherwise. (a) Explain why a, hence b, must be nonnegative. Graph the function f. (b) Find the value of b'-a. 7 (c) If it is known that P(X 8).Suppose that the random variable X has the probability density function for -1s x < 1 c(1– x?) f(x) = elsewhere Find the value of the constant c.(47) Let X z b(8,-) find E(5+6x) and distribution function.
- * Consider the probability density function fx (x) = a e-b lel where X is the random %3D variable which assumes all the values from (i) relation between a and b -00 to -00, Find (ii) the probability of finding X in the range 1 to 2.10) Let f(x) by the probability density function of some continuous random variable. Explain why it must be true that f(x) dx =1.Let the density function of a random variable X be given by fx(x) = 0x®-', 0A random variable X has the probability density function 1 1 f(x) πx2 + 1 ' then the probability that X2 lies between and 1 is:Figure I shows the piecewise function (I), (II), (III) and (IV) for cumulative distribution function F(x) for continuous random variable. F(x) (6. 1) IV III II (4.0.8333) (0.0.1667) Figure I Construct the probability density function fix). Should one of the piecewise functions (IV) is not constant, explain the changes.Let X be a random variable with pdf f(x) = kx*,-1Consider the three functions I. f(x)= {1-x if 0 < x < 2 {0 otherwise II. f(x)= {1+x if 0 < x < 2 {0 otherwise III. f(x)= {(3/27)x^2) if 0< x < 3) {0 otherwise Which of these above functions can be probability distributions? And why?The service life of an item is defined as an item's total life in use from the point of sale to the point of discard. A probability density function often used to model the service life of items is of the form k-1 Pk (x) = 4/1 (1) * ¹.-(-)* where x is measured in years for x ≥0, and k is a constant such that k 21. The value of k is determined by the failure rate of the item over time. (a) Consider the case where k = 2, resulting in the probability density function P²(x) = 20 (² The function p₂(x) can be used to model the service life of Product A. (i) Using definite integrals, determine the probability that a randomly selected Product A will have a service life of 1.5 years or less. (ii) An approximate measure of the anticipated service life for Product A is m, the median se life of the item, which can be calculated using the equation: P₁(x)dx=0.5. Using your answer to part (a)(i), explain why the median service life of Product A is greater than 1.5 years. page 12 of 15Q2) The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the below density function , Find the probability that a bottle of this medicine will have a shelf life of anywhere between 60 and 120 days 20000 0SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,
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