Suppose that X is a non-negative random variable with continuous density function fx (z) and a > 0. Show that P(X > a) ≤ ² E(X)
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- Suppose that X₁, ..., X₁ are i.i.d. random variables with density function ƒ(x|0) = e−(x−0), x ≥ 0 and ƒ(x|0) = 0 otherwise. a) Find the method of moments estimator 0 b) Find the MLE of 0Let X and Y be continuous random variables with joint probability density function f(x, y) = (3/200y if 0 < 5x < y < 10 O otherwise Find Cov(X, Y)Let X be a continuous random variable with density function be-bx for a > 0 f (x) = otherwise where b > 0. - find M(t) the moment generating function of X, then what is E(x) ?
- Suppose that X and Y are random variables with the joint density function cx² + cy, 0 sx< 3,1Consider two continuous random variables X and Y with marginal distributions g(x) and h(y)respectively and the joint density function given by: x > 0, 0 < y < 2 elsewhere. f (r.y) Then: X and Y ar statistically dependent f(x,y)#g(x)h(y) None of these f(ylx)=g(x)For some c, a continuous random variable X has density f(x) = { ' 0, (a.) What is c? (b.) Find ux and o. for 1≤x≤4 otherwiseSuppose X is a continuous random variable with density f(x) = x/2 , 0 <= x <=2 f(x) = 0 , elsewhere Write an integral expression for the moment generating function M(t).Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0Suppose that two-dimensional continuous random variable (X, Y) has joint probability density function given by f(x,y) = 24xy, x is less than equal to 1 and greater than equal to 0, y is less than equal to 1 and greater than equal to 0, x+y is less than equal to 1 and greater than equal to 0. Check that E(Y) = E[E(Y|X)] and V(Y) = E[V(Y|X)] + V[E(Y|X)].Let x be a continuous random variable with density function f (x) = { a*e-* for x>0 0, el sewhere where b > 0. Calculate the mode of X andLet Y1,..., Yn denote a random sample from the density function given by fy (yla, 0) = F(a)0aY"e3 for y >0, where a > 0 is known and r(-) is the gamma function. Find the MLE of 0.6. Let X and Y be continuous random variables with joint density function 24xy if 0 < x < 1,0 < y < 1 – x f (x, y) = 0. otherwise. Calculate E(Y|X = }).Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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