Question 3.3: Let X be a r.v. with p.d.f. f (x) = 3x², 0 < x < 1. Th......
Let X be a r.v. with p.d.f. f(x)=3x^{2},\ 0\lt x\lt 1. Then:
(i) Calculate the quantities: E X,E X^{2}, and V\!\!ar(X).
(ii) If the r.v. Y is defined by: Y=3X-2, calculate the EY and the V\!\!ar(Y).
(i) By (3),
E X=\textstyle\int_{0}^{1}x\cdot3x^{2}\,d x={\textstyle\frac{3}{4}}x^{4}|_{0}^{1}={\textstyle\frac{3}{4}}=0.75, whereas by (7),
E X^{k}=\sum\limits_{i=1}^{n}{x_{i}^{k}}f(x_{i})\quad{\mathrm{or}}\quad E X^{k}=\sum\limits_{i=1}^{\infty}{x_{i}^{k}}f(x_{i})\quad{\mathrm{or}}\quad E X^{k}=\int_{-\infty}^{\infty}{x^{k}}f(x)\,d x.\qquad (7)
applied with k = 2, E X^{2}=\int_{0}^{1}x^{2}\cdot3x^{2}\,d x={\frac{3}{5}}\,=\,0.60, so that, by (9),
V\!\!ar(X)=0.60-(0.75)^{2}=0.0375.
(ii) By (4) and (6),
E(c X)=c E X,\quad E(c X+d)=c E X+d,\quad{\mathrm{where~c~and}}\,d\,\mathrm{are~constants}.\quad(4)
E Y=E(3X-2)=3E X-2=3\times0.75-2=0.25, whereas by (10),