Probability MT22, Random variables

Created: June 21, 2023 | Updated: October 21, 2025 | About these notes | View in context | Study these flashcards

Flashcards

@State the definition for a _ discrete _ random variable, and then the definition of just a random variable?

$X$ is a discrete random variable if

  • $\text{Im}(X)$ is a finite or countable subset of $\mathbb R$
  • $\forall x \in \text{Im}(X), \{\omega : X(\omega) = x\} \in \mathcal F$

$X$ is a random variable if

@todo.

If $X _ 1, X _ 2, X _ 3$ are identically distributed and independent, are $X _ 1 + X _ 2 + X _ 3$ and $3X _ 1$ the same?

No.

For an event $A$, what is the definition of the random variable $\mathbb{1} _ A$?
\[\mathbb{1} _ A = \begin{cases} 1 \text{ if } A _ i \text{ occurs} \\\\0 \text{ otherwise} \end{cases}\]

Discrete random variables

What sets is a discrete random variable $X$ a function on?
\[\Omega \to \mathbb{R}\]

@State the two conditions for function $p _ X(x)$ to be a probability mass function.
  • $\forall x: p _ X(x) \ge 0$
  • $\sum _ {x\in\text{Im }X} p _ X(x) = 1$

What is the first condition for a function $X : \Omega \to \mathbb{R}$ to be a discrete random variable, in English (it’s about images)?

The image of $X$ is a countable set.

What is the second condition for a function $X : \Omega \to \mathbb{R}$ to be a discrete random variable (about what it means for $X = x$)?
\[\\{\omega \in \Omega : X(\omega) = x\\} \in \mathcal{F}\]

Can you expand “the image of a discrete random variable $X$” into notation?
\[\\{X(\omega) : \omega \in \Omega\\}\]

What is the probability mass function of a discrete random variable $X$?
\[p _ X(x) = \mathbb{P}(X = x)\]

Given a function $p _ X(x)$ that satisfies the conditions for being a probability mass function, how could we define a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a random variable $X$ that corresponds to this probability mass function?
  • $\Omega = \{x \in \mathbb{R} : p(x) \ne 0\}$
  • $\mathcal{F} = \mathcal{P}(\Omega)$
  • $\mathbb{P}(S) = \sum _ {\omega \in S} p(\omega)$
  • $X(\omega) = \omega$

Suppose B is an event such that $\mathbb{P}(B) \ne 0$. What is the definition of the “conditional mass function of X given B”?
\[\mathbb{P}(X = x \vert B) = \frac{\mathbb{P}(\\{X = x\\} \cap B)}{\mathbb{P}(B)}\]

What is $\mathbb{E}[h(X, Y)]$ for discrete random variables $X$ and $Y$?
\[\sum _ {x \in \text{Im }X} \sum _ {y \in \text{Im }Y} h(x, y) \mathbb{P}(X = x, Y = y)\]

What does $\text{Im} X$ mean when talking about a discrete random variable?

The set of all values that $X$ can take.

How could you calculate $\mathbb{P}(X = x \vert B)$?
\[\frac{\mathbb{P}(X=x, B)}{\mathbb{P}(B)}\]

Jointly distributed discrete random variables

What must hold about any joint probability mass function $p _ {X, Y}(x, y)$?
\[\sum _ {x\in\text{Im}X} \sum _ {y\in\text{Im}Y} p _ {X,Y}(x, y) = 1\]

How can you determine the marginal distribution $\mathbb{P}(X = x)$ given the joint distribution $\mathbb{P}(X = x, Y = y)$?
\[\mathbb{P}(X = x) = \sum _ {y \in \text{Im} Y} \mathbb{P}(X = x, Y = y)\]

What is true about the joint distribution $p _ {X,Y}(x,y)$ and the marginal distributions $p _ X(x)$ and $p _ Y(y)$ when $X$ and $Y$ are independent?
\[p _ {X,Y}(x, y) = p _ X(x) p _ Y(y)\]

How could you write the definition of the CDF $F _ X$ in the discrete case?
\[F _ X(x) = \sum _ {u\in(-\infty, x] \cap \text{Im}X} p _ X(u)\]

Continuous random variables

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.

Given a continuous random variable $X$ with PDF $f _ X(x)$, and $Y = aX + b$, what’s the first step in determining the PDF $f _ Y(y)$?

Rewriting the CDF in terms of $X$:

\[F _ Y(y) = F _ X\left(\frac{y-b}{a}\right)\]

You have a continuous random variable $X$ with PDF $f _ X(x)$, and $Y = aX + b$. You’ve written the CDF in terms of $X$ like $F _ Y(y) = F _ X\left(\frac{y - b}{a}\right)$. How do you now determine $f _ Y(y)$?

Differentiate $F _ X\left(\frac{y-b}{a}\right)$ with respect to $y$.

@Prove that if $F _ X$ is the cumulative density function then

  1. $F _ X$ is non-decreasing
  2. $\mathbb P(a < X \le b) = F _ X(b) - F _ X(a)$
  3. As $x \to \infty$, $F _ X(x) \to 1$.
  4. As $x \to -\infty$, $F _ X(x) \to 0$.

@todo (probability, page 47).

@important~

Suppose that $X$ is a continuous random variable with density $f _ X$ and that $h : \mathbb R \to \mathbb R$ is a strictly increasing differentiable function. @Prove that $Y = h(X)$ is a continuous random variable with p.d.f.

\[f _ Y(y) = f _ X(h^{-1}(y))\frac{\text d}{\text d y} h^{-1}(y)\]

@todo (probability, page 56).

@important~

Jointly distributed continuous random variables

For two continuous random variables $X, Y$ then what is $F _ {X,Y}(x,y)$ shorthand for?
\[\mathbb{P}(X \le x, Y \le y)\]

What form does the joint CDF $F _ {X,Y}(x,y)$ have to be for $X$ and $Y$ to be jointly continuously distributed with PDF $f _ {X,Y}$?
\[F _ {X,Y}(x,y) = \int^x _ {-\infty}\int^y _ {-\infty} f _ {X,Y}(u,v)\text{d}v\text{d}u\]

For a jointly continuously distributed $(X, Y)$, what is $\mathbb{P}(a < X \le b, c < Y \le d)$ in terms of the joint CDF?
\[F _ {X,Y}(b, d) - F _ {X,Y}(a,d) - F _ {X,Y}(b,c) + F _ {X,Y}(a,c)\]

For “nice enough” $A \subseteq \mathbb{R}^2$, what is $\mathbb{P}((X,Y) \in A)$?
\[\iint _ {(x,y) \in A} f _ {X,Y}(x,y) \text{d}y\text{d}x\]

If $X, Y$ are jointly continuous with joint density function $f _ {X,Y}$, how can you recover $f _ X(x)$?
\[f _ X(x) = \int^\infty _ {-\infty} f _ {X,Y}(x,y) \text{d}y\]

If $X, Y$ are jointly continuous with joint density function $f _ {X,Y}$, how can you recover $f _ Y(x)$?
\[f _ Y(x) = \int^\infty _ {-\infty} f _ {X,Y}(x,y) \text{d}x\]

What does it mean for continuous random variables $X$ and $Y$ to be independent?

If $f _ {X, Y} (x, y) = f _ X(x) f _ Y(y)$ for all $x$ and $y$.

How can you rewrite $F _ {X,Y}(x,y)$ for independent continuous random variables $x$ and $y$?
\[F _ {X,Y}(x, y) = F _ X(x) F _ Y(y)\]

When you two random variables whose domain are $(0, 1)$, then what do probabilities correspond to?

Areas.

Suppose $X$ and $Y$ are jointly continuous random variables. @Prove that

\[\mathbb P(a < X \le b, c < Y \le d) = \int^d _ c \int^b _ a f _ {X,Y}(x, y) \text d x \text d y\]

@todo (probability, page 58).

@Prove that if $X$ and $Y$ are jointly continuous with joint density $f _ {X, Y}$ then $X$ is a continuous random variable with density

\[f _ X(x) = \int^\infty _ {-\infty} f _ {X,Y} (x, y) \text d y\]

and likewise

\[f _ Y(y) = \int^\infty _ {-\infty} f _ {X,Y} (x, y) \text d x\]

@todo (probability, page 59).



Related posts