Question

3. Let X have density f_X(x) = C/sqrt(x), 0 < x < 1. (a) What is the value of C? (b) Compute P(1/4 < X < 25/16). (c) Find the cumulative distribution F_Y(y) and probability density function f_Y(y) of Y = X(1 - X). (d) Find probability density function f_Z(z) of Z = X^{1/4}. (e) Find the mean and variance of X. (f) Calculate the expected value of Z by (i) evaluating integral from -infinity to infinity psi(x)f_X(x)dx for an appropriate function psi(x), (ii) evaluating integral from -infinity to infinity zf_Z(z)dz, (iii) approximation using an appropriate formula based on Taylor series expansion of x^{1/4}. (g) Calculate the variance of Z and compare it to the approximation using an appropriate formula based on Taylor series expansion of x^{1/4}.

          3. Let X have density
f_X(x) = C/sqrt(x), 0 < x < 1.
(a) What is the value of C?
(b) Compute P(1/4 < X < 25/16).
(c) Find the cumulative distribution F_Y(y) and probability density function f_Y(y) of Y = X(1 - X).
(d) Find probability density function f_Z(z) of Z = X^{1/4}.
(e) Find the mean and variance of X.
(f) Calculate the expected value of Z by
(i) evaluating integral from -infinity to infinity psi(x)f_X(x)dx for an appropriate function psi(x),
(ii) evaluating integral from -infinity to infinity zf_Z(z)dz,
(iii) approximation using an appropriate formula based on Taylor series expansion of x^{1/4}.
(g) Calculate the variance of Z and compare it to the approximation using an appropriate formula based on Taylor series expansion of x^{1/4}.

3. Let X have density
fX(x) = C/sqrt(x), 0 < x < 1.
(a) What is the value of C?
(b) Compute P(1/4 < X < 25/16).
(c) Find the cumulative distribution FY(y) and probability density function fY(y) of Y = X(1 - X).
(d) Find probability density function fZ(z) of Z = X^1/4.
(e) Find the mean and variance of X.
(f) Calculate the expected value of Z by
(i) evaluating integral from -infinity to infinity psi(x)fX(x)dx for an appropriate function psi(x),
(ii) evaluating integral from -infinity to infinity zfZ(z)dz,
(iii) approximation using an appropriate formula based on Taylor series expansion of x^1/4.
(g) Calculate the variance of Z and compare it to the approximation using an appropriate formula based on Taylor series expansion of x^1/4.

Added by Bego-A V.

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