Lecture - Probability MT22, XII

What is the only condition for $X: \Omega \to \mathbb{R}$ being a random variable?

\[\\{w \in \Omega : X(\omega) \le x\\} \in \mathcal{F}\]

What is the formula for the cumulative distribution function (CDF) $F _ X : \mathbb{R} \to \mathbb{R}$ of a random variable?

\[F_X(x) = \mathbb{P}(X \le x)\]

What is true about the monotonicity of the CDF?

It is monotonically increasing.

What is $\lim _ {x \to \infty} F _ X(x)$?

\[1\]

What is $\lim _ {x \to -\infty} F _ X(x)$?

\[0\]

How can you calculate (for $a < b$) $\mathbb{P}(a < x \le b)$ using the CDF $F _ X(x)$?

\[F_X(b) - F_X(a)\]

What’s the very specific inequality for $F _ X(b) - F _ X(a)$?

\[a < x \le b\]

How is the probability density function related to the cumulative distribution function $F _ X$ for a continuous random variable?

\[F_X(x) = \int^x_{-\infty} f_x(u) \text{du}\]

What is the condition for a random variable to be a continuous random variable?

It can be written in terms of an integral.

For any continuous random variable, what is $\mathbb{P}(X = x)$?

\[0\]

Why doesn’t the fact that

\[\lim_{n \to \infty} \mathbb{P}(A_n) = \mathbb{P}\left(\bigcup^\infty_{n = 1} A_n\right)\]

for an increasing family of events contradict the fact that $\mathbb{P}(X = x) = 0$ for a continuous random variable?

Because the above only works for countable unions.