Can anyone please help me with this random variable question I've stumbled across.
Recall from calculus that a function $h$ is called non-decreasing if $x\leq y$ implies $h(x)\leq h(y)$, for every $x,y \in \textrm{dom}(h)$.
Q1a) Let $X$ be a continuous random variable with probability density function $f$. Prove that the probability distribution function of $X$. i.e. $F(x)=\int_{-\infty}^x f(y)dy$, is a non decreasing function of $x$ that belongs to $R$.
Q1b) Show that $\displaystyle\lim_{x\to-\infty}F(x)=0$ and $\displaystyle\lim_{x\to+\infty}F(x)=1$, and explain the probabilistic meaning of these facts
Best Answer
Hint for Q1a: The cumulative distribution function (CDF), by definition, has the property that $P(X \leq x) = F(X)$. So you'll have to ponder whether it's possible that for $x_1 \leq x_2$, $P(X \leq x_1) > P(X \leq x_2)$. What would that mean for the probability $P(x_1 < X \leq x_2)$?
Hint for Q1b: If $X$ is a real random variable, what is $P(X \in \mathbb{R})$?