Question

Assume that X has a uniform distribution over [0; 1] and that Y has the uniform distribution over

[2; 3]: Which of the following statements are true and which are false? Justify your answers!

(a) P (X < Y) = 1.

(b) Since X is smaller than Y , P (X < 1) > P (Y < 1).

(c) There are some values a for which P (X < a) = P (Y < a).

Answer 1

(a) The first statement is correct. Suppose that $X,Y:\Omega\rightarrow\mathbb{R}$. Denote by $\Omega_1\sub\Omega$ a subset of $\Omega$ such that $X(w)\notin[0,1]$ for all $w\in\Omega_1$. Denote by $\Omega_2\sub\Omega$ a subset of $\Omega$ such that $Y(w)\notin[2,3]$, for all $w\in\Omega_2$. Denote by $\Omega_3$ a subset of $\Omega$ satisfying $X(w)<Y(w)$ for all $w\in\Omega_3$. It is clear that $\Omega_3\sub\Omega_1\cup\Omega_2$. But $P(\Omega_1\cup\Omega_2)\leq P(\Omega_1)+P(\Omega_2)=0$. We receive that $P(X<Y)=P(\Omega_3)=0$

(b) $P(Y<1)=0$, since $Y$ has the uniform distribution over $[2,3]$. $P(X<1)=1$. We receive: $P(X<1)>P(Y<1).$

(c) We can take $a=0$. We get $0=P(X<0)=P(Y<0)$. due to the definition of the uniform distribution.