Question

. [19 points] Refer to Problem 1. (a) Find the cumulative distribution function F(x). Be sure to write your answer in the appropriate way. [4 points] (b) Find the mean value of X, i.e. find E(X). [4 points] (c) Find the second moment, i.e. find E(X 2 ). [4 points]

Answer 1

Answer:

Step-by-step explanation:

a) We know that the cummulative distribution function of a continous random variable X is defined as:

$F(x)=\int\limits^x_{-\infty} {f(t)} \, dt$

given: $f(x)=-\frac{2}{25} x + \frac{2}{5}$

hence its cdf will be

$F(x)=\int\limits^x_0 (-\frac{2}{25}t + \frac{2}{5} )\, dt\\\\=\frac{2}{5}[-\frac{t^2}{10} + t]\limits^x_0 \\\\F(x)=-\frac{x^2}{25} +\frac{2}{5}x$

b) Mean of X is given as:

$E(x)=\int\limits^{\infty}_{-\infty} {f(x)} \, dx\\\\=\int\limits^{5}_{0} (-\frac{2}{25}x+\frac{2}{5})x \, dx\\\\\frac{2}{5}\int\limits^{5}_{0} (-\frac{1}{5}x^2+x) \, dx\\\\=\frac{2}{5}(-\frac{x^3}{15}+\frac{x^2}{2})\limits^5_0\\\\E(x)=\frac{5}{3}$

c) For second moment

$E(x^2)=?\\\\E(x^2)=int\limits^{\infty}_{-\infty} {x^2f(x)} \, dx\\\\=\int\limits^{5}_{0} {(-\frac{2}{25}x + \frac{2}{5})} \, dx\\\\=\frac{25}{6}$