Y =)Z (n>2) Σz i=1 ) Express the moment generating function My(t) of the random variable Y in terms of moment generating functions involving the random variables Zi, i = 1,..., n. ›) Determine My(t) for the special case that Z₂ ~ N(0, 1). 2) For the above special case, calculate E[Y] by using the moment generating function.
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Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variables
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- Suppose that Z₁, Z2, ..., Zn are statistically independent random variables. Define Y as the sum of squares of these random variables: n Y=)Z (n>2) i=1 (a) Express the moment generating function My(t) of the random variable Y in terms of moment generating functions involving the random variables Z, i = 1, ..., .., n. (b) Determine My(t) for the special case that Z₁ ~ N(0, 1). (c) For the above special case, calculate E[Y] by using the moment generating function.Let X be a continuous random variable whose moment generating function is 4x(t) = (5) ³. Find Var (X)d) Let Y be a continuous random variable with a probability mass function: - 1 fV) = (,) p"(1– p)"-* - 1 Derive the moment generating function (MGF) of a random variable Y.
- Suppose the random variable X~ Beta(a, 3), namely its PDF fx(x) = I(a + 3) r(a)(3) -ra-1(1−z)3-1, 0 < x < 1, Beta(a +3, y), namely its PDF fy (y) = I(a +8+y) r(a + 3)(y) ¹(1 − y)*-¹, 0 < y < 1, and X and Y are independent. Define U = XY, V = X. Find the joint PDF fuv(u, v).Q3/(a) Let X = b(n,) and E(x)= E(16-7x) find n. (b)Let X be a random variable with p.d.f, ƒ(x)= c 3 x=1,2,3,.., zero elsewhere. Find the constant c.Q3/(a) Let X = b(n, ) and E(x)=E(16–7x) find n. (b)Let X be a random variable with p.d.f, f(x)=c x =1,2,3,.., zero elsewhere. Find the constant c.
- Let random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Moment generating function is given: Mx(t)= 1/(4-t)^2 Find the mean and the vairance for the random variable x.
- b) Let Z₁-N(0,1), and W₁ = Y~N(0,1), for i=1,2,3,...,10, then: dx dy i) State, with parameter(s), the probability distribution of the statistic, T = - 154 ii) Find the mean and variance of the statistic T = ₁² 10 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ß such that P(T> B) = 0.01, where T = ₁2₁² +².Let Mx (t) = 1/(1-t), t < 1 be the moment-generating function of a random variable X. Find the moment-generating function of the random variable Y = 2X +1.(b) (i) Briefly explain how moment generating function of a random variable, y may be used to Generate its moments. (ii) Let the random variable, y have probability density function S(t) = kye-1/2y y20 , elsewhere Where k> 0 is a constant. Find the moment generating function of y and use it to evaluate the mean and variance of y leaving your results in terms of k.