a) Suppose that, for-1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by f(x₁,x₂) = {(1 ([1-a(1-2e-x¹)(1-2e-x2)Je*1-*2,0 ≤ x₁,0 ≤ x₂. otherwise i) Find the marginal distribution of X₁. ii) Find E(X₁X₂).
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![a) Suppose that, for -1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by
f(x₁,x₂) = {[1-a(1-
([1-a(1-2e-x1)(1-2e-x2)]e*1-X2
,0 ≤ x₁,0 ≤ x₂.
0,
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%2F98045eb8-67c9-4c5f-b511-4afdd76e7951%2F40c217cd-dec4-4d1c-afd2-3480371d8e12%2Fyutk5t7_processed.jpeg&w=3840&q=75)
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- The probability density function of the continuous random variable X defined in the set of non-negative real numbers is given as f(x) = 2.exp(-2x). What is the expected value of X?The probability density of the random variable Z isgiven by f(z) = kze−z2for z > 00 for z F 0Find k and draw the graph of this probability density.Sketch the graph of the probability density function over the indicated interval. f(x) = 2x/25, [0, 5] Find the indicated probabilities. a) P(0 < x < 3) b) P(1 < x < 3) c) P(3 < x < 5) d) P(x ≥ 1)
- Sketch the graph of the probability density function over the indicated interval and find the indicated probabilities. f(x)= x/50, [0, 10] A) P(0<x<6) B)P(4<x<6) C)P(3<x<5) D)P(x≥1)Q2) 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 0d) Consider the continuous random variable with probability density function f(x) = {1t. for Ost≤1 -t. for-1 st ≤0 Estimate the standard deviation of the random variable.Let X1,., X, denote a random sample from the probability density function f(r) = 0xº-1, 0 0. Show that X is a consistent estimator of .The function f(x) is the probability density function, that describes the random variable travel time X. f(x) = {xe3,_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?Let X be the random variable having pdf f(x) = 1,0 < x < 1, zero e.w. Compute the probability of the range of the random sample of size 4 (X1, ., X4) is less than 1/2. Hint: Range is R = X(4) – X(1).Suppose that therandom variable X has the probability density function c(1– x2) for -1s x <1 f(x) elsewhere What is the variance of XLet X₁, X2, X3, X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x) = (0, -, if 0Suppose that the probability density function of x is (3x², 0 (2/3)) b) Determine the cumulative distribution function of x.SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,