If y is a random variable with a mass function, its probability is defined as: y 0 1 4 9 P(y) 0.64 0.25 0.09 0.02 find E(√) E(y² + 1) ) Y
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- A statistician has to decide on the basis of a singleobservation whether the parameter θ of the density f(x) =⎧⎨⎩2xθ 2 for 0 < x < θ0 elsewhere equals θ1 or θ2, where θ1 < θ2. If he decides on θ1 when theobserved value is less than the constant k, on θ2 when theobserved value is greater than or equal to the constant k,and he is fined C dollars for making the wrong decision,which value of k will minimize the maximum risk?Suppose 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).A cylinder of radius r and height h has volume V = nr 2h.(a) Use the Linear Approximation to show that (b) Estimate the percentage increase in V if r and h are each increasedby 2%.
- Find the normal line to f(x)=ax^2-3ax at x=2. Assume that a is a positive constant.A pulse (gas) travelling along a fine straight uniform tube filled with gas causes the density at any time t and distance x from the origin where the velocity is u_{0} to become roh_{0}ϕ(vt-x). Prove that the velocity u (at time t and distance x from the origin) is given bySuppose that the random variables X and Y have a joint density function f(x,y).prove that Cov(X,Y)=0 if E(X|Y=y) does not depend on y
- Find the moment generating function for the random variable X whose density function is f(x). fx = 2x 0<x<1 MX(t) = _______________________________________Suppose a continuous random variable X~Fx(x): f(x,y) = {1/4e^-1x/4, if x≥0 0, x<0} What is the cumulative density function of Y=min{2,X}?If X is exponentially distributed with parameter λ and Y is uniformly distributed on the interval [a, b], what is the moment generating function of X + 2Y ?
- A rectangular plate with insulated surface is 10 cm. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u(x,0) = 8 sin(px/ 10) when 0 <x <10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0o C, find the steady state temperature distribution u(x,y).For x1, the marginal distribution fx1 (x1) function is validis it a density function)Consider the seriesxt = sin(2πU t),t = 1, 2, . . ., where U has a uniform distribution on the interval (0, 1). Prove xt is weakly stationary
