4.5-4. Let X and Y have a bivariate normal distribution with μx Find = = 75, µy = 5, σ = 144, σ = 225, and p = 0.6. με (a) E(YX=76). (b) Var(YX = 76). (c) P(Y≤ 90, |X = 76). 'Χ
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- The variable X has a normal distribution with mean u = 100 and standard deviation o = 8. The variable Y has a normal distribution with mean u = 85 and standard deviation o = 6. Variables X and Y are independent. (a) Find the mean, u, and standard deviation, o, of the sum X + Y. = = (b) Find the mean, u, and standard deviation, a, of the difference X - Y. O =Let X be a random variable normally distributed with mean μx 1. Find the mean μy of Y. 2. Find the standard deviation oy of Y. 3. Find the PDF fy of Y and evaluate fy (5). (My, oy, fy (5)) = ___________). = 6 and standard deviation ox = 4. Let Y = 4X + 5.5. Show that for a normal distribution with mean µ 2 and variance o', the central moments satisfy the relation. Hak+1 = 0 and pu, = (2K-1)u,².
- X1,X2,...,Xn are i.i.d. random variables from N(0,θ) distribution. Here, θ= σ2 >0is the variance of the distribution, and is an unknown parameter. Find the MLE of θ(no need tocalculate the second derivative).The moment generating function can be used to find the mean and variance of the normal distribution. Use derivatives of Mx(t) to verify that E(X) = µ and V(X) = 0².Let X and Y have a bivariate normal distribution with mean vector (μ₁ = 3, μ2 = 1)′, and standard deviations ₁ = 4,02 = 5, and correlation p = 0.6. (i.) Find E(Y|X= 7) (ii.) Find Var(Y|X = 7) (iii) Find P(3The random variable E is described by the distribution (A · ]sin(e)|, fE(e) = f(x) = {" -TT4. The annual precipitation X in a city is modeled by a normal distribution with a mean value of ux = 125 cm and a coefficient of variation of dx = 0.2. Determine the following: a) The standard deviation of X. b) Р(X150 cm) d) P(100 ст <Х< 140 ст) e) The value xo such that the probability of the annual precipitation exceeding xo is only 1/3 of that of not exceeding xo.Recommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage