Lectures: CSE206_Fa24-03.pdf Lab/Tutorial: week03.pdf
- This week introduces random variables as functions on a probability space that produce numerical outcomes.
- We define indicator random variables, which connect events and random variables and are modeled as Bernoulli variables.
- Every random variable has a distribution function (cdf), and discrete ones also have a probability mass function (pmf).
- We focus on discrete random variables and key examples: Bernoulli, Binomial, discrete uniform, Geometric, and Poisson.
- Independence is extended from events to random variables, including equivalent conditions in terms of pmfs.
- Random vectors (several random variables together) are introduced with joint, marginal, and conditional pmfs.
- The lab builds practice in working with joint distributions, transforming random variables, and recognizing standard discrete distributions.
- Plain-language definition.
- A random variable is a numerical quantity whose value depends on the outcome of a random experiment.
- Formally it is a function defined on the sample space, but usually we think of it just as “the number we measure.”
- Formal definition.
- Given a sample space
$\Omega$ with σ-algebra$\mathcal{F}$ , a random variable$X$ is a function $$ X:\Omega \to \mathbb{R} $$ such that for every real number$x$ , the set${\omega\in\Omega : X(\omega)\le x}$ is in$\mathcal{F}$ .
- Given a sample space
- Intuition / mental model.
- The function
$X$ “reads” the outcome$\omega$ and returns a number (e.g., number of heads, total score, waiting time). - The measurability condition (“${X\le x}\in\mathcal{F}$”) just says “events described in terms of
$X$ ” are legitimate events.
- The function
- Tiny example.
- Toss a fair coin twice, with
$\Omega = {HH, HT, TH, TT}$ . - Let
$X$ be “number of heads”:$X(HH)=2$ ,$X(HT)=1$ ,$X(TH)=1$ ,$X(TT)=0$ . This is a random variable.
- Toss a fair coin twice, with
- Plain-language definition.
- An indicator of an event
$E$ is a random variable that is 1 when the event happens and 0 otherwise.
- An indicator of an event
- Formal definition.
- For
$E \in \mathcal{F}$ , the indicator function$I(E)$ is $$ I(E)(\omega) = \begin{cases} 1, & \omega\in E,\ 0, & \omega\notin E. \end{cases} $$
- For
- Intuition / mental model.
- Indicators allow us to translate event statements into algebra with random variables (sums and products).
- Many combinatorial probability problems can be rewritten using indicator variables and simplified.
- Tiny example.
- In the network example from lecture 2, indicator variables
$I(E_1),\dots,I(E_4)$ for working channels are used; the event “there is communication from$N_1$ to$N_3$ ” is $$ (I(E_1)+I(E_2))(I(E_3)+I(E_4)) > 0. $$ - Any indicator random variable is a Bernoulli random variable (see 2.6.1).
- In the network example from lecture 2, indicator variables
- Plain-language definition.
- The distribution function (also called cumulative distribution function, cdf) of a random variable
$X$ tells us, for each real number$x$ , the probability that$X$ is at most$x$ .
- The distribution function (also called cumulative distribution function, cdf) of a random variable
- Formal definition.
- For a random variable
$X$ on a probability space$(\Omega,\mathcal{F},P)$ , its cdf is $$ F_X(x) = P(X \le x), \quad x\in\mathbb{R}. $$
- For a random variable
- Intuition / mental model.
-
$F_X(x)$ accumulates probability mass as$x$ increases; plotting$F_X$ shows how probability is distributed along the real line. - For discrete
$X$ , the cdf is a step function; for continuous$X$ (later), it will be continuous and usually smooth. - Example cdf shape from lecture:
-
- Basic properties (from lecture).
-
$\lim_{x\to -\infty}F_X(x)=0$ . -
$\lim_{x\to +\infty}F_X(x)=1$ . -
$F_X$ is non-decreasing (never goes down as$x$ increases). -
$F_X$ is right-continuous (no “jumps” from the right).
-
- Tiny example.
- For
$X$ = number of heads in two tosses of a fair coin, possible values are 0, 1, 2, and the lecture gives $$ F_X(x)=\begin{cases} 0, & x<0,\ 1/4, & 0\le x<1,\ 3/4, & 1\le x<2,\ 1, & x\ge 2. \end{cases} $$
- For
- Plain-language definition.
- A random variable is discrete if it takes values only in a countable set (finite or countably infinite), and all probability is concentrated on those values.
- Formal definition.
-
$X$ is discrete if there exists a countable set$C\subset\mathbb{R}$ such that$P(X\in C)=1$ . - Its probability mass function (pmf) is $$ p_X(x)=P(X=x),\quad x\in\mathbb{R}. $$
- For a discrete random variable,
$$
\sum_{x\in\mathbb{R}} p_X(x) = 1,
$$
and since
$X$ is discrete, only countably many terms are non-zero.
-
- Intuition / mental model.
- Discrete variables are used for counts (number of visitors, tosses until first tails, number of successes, etc.).
- The pmf is a “table” of probabilities at points; the cdf is the running sum of those probabilities up to
$x$ .
- Relationship between pmf and cdf (for discrete).
- If the values of a discrete
$X$ with positive probability can be ordered$x_1<x_2<x_3<\dots$ , then $$ p_X(x_j) = F_X(x_j) - \lim_{x\to x_j^-}F_X(x). $$ - So the size of the jump of the cdf at
$x_j$ equals the pmf value at$x_j$ .
- If the values of a discrete
- Tiny example.
- For
$X$ = number of heads in two tosses: $$ p_X(0)=1/4,\quad p_X(1)=1/2,\quad p_X(2)=1/4,\quad p_X(x)=0 \text{ otherwise}, $$ and the cdf is as in 2.3.
- For
- Plain-language definition.
- Several random variables are independent if knowing the values of some of them does not change the probability distribution of the others.
- Formal definition.
- Random variables
$\xi_1,\dots,\xi_n$ on the same probability space are independent if for all real numbers$x_1,\dots,x_n$ : $$ P(\xi_1\le x_1,\dots,\xi_n\le x_n) = F_{\xi_1}(x_1)\cdots F_{\xi_n}(x_n). $$ - For discrete random variables, this is equivalent to saying that for all
$x_1,\dots,x_n$ $$ P(\xi_1 = x_1,\dots,\xi_n = x_n) = p_{\xi_1}(x_1)\cdots p_{\xi_n}(x_n), $$ and equivalently that events${\xi_1=x_1},\dots,{\xi_n=x_n}$ are independent.
- Random variables
- Intuition / mental model.
- Independence extends the idea “$P(A\cap B)=P(A)P(B)$” to statements about multiple random variables.
- For discrete variables, the joint pmf factors into the product of the individual pmfs.
- Tiny example.
- Independent Bernoulli trials: if
$\xi_1,\dots,\xi_n$ are independent$\text{Bernoulli}(p)$ variables, then for any pattern$(x_1,\dots,x_n)\in{0,1}^n$ , $$ P(\xi_1=x_1,\dots,\xi_n=x_n) = \prod_{j=1}^n p^{x_j}(1-p)^{1-x_j}. $$
- Independent Bernoulli trials: if
- Definition.
- A random variable
$\xi$ is Bernoulli with parameter$p$ if $$ P(\xi=1)=p,\quad P(\xi=0)=1-p, $$ usually with$q=1-p$ . We write$\xi\sim \text{Bernoulli}(p)$ .
- A random variable
- Intuition.
- Models the outcome of a single “yes/no” experiment: success (1) or failure (0).
- Indicator random variables are Bernoulli.
- Tiny example.
- Toss a biased coin with probability
$p$ of heads. Let$\xi = I(\text{heads})$ . Then$\xi\sim\text{Bernoulli}(p)$ .
- Toss a biased coin with probability
- Definition.
- A random variable
$\xi$ is binomial with parameters$n$ and$p$ , written$\xi\sim\text{Bin}(n,p)$ , if its pmf is $$ p_\xi(x) = \binom{n}{x} p^x(1-p)^{n-x} I(x\in{0,\dots,n}). $$
- A random variable
- Intuition.
- Models the number of successes in
$n$ independent Bernoulli$(p)$ trials. - If
$\xi_1,\dots,\xi_n$ are independent$\text{Bernoulli}(p)$ and$\xi = \xi_1+\dots+\xi_n$ , then$\xi\sim\text{Bin}(n,p)$ (this is stated and will be proved in the next lecture).
- Models the number of successes in
- Tiny example.
- Toss a coin
$n=5$ times with success probability$p$ . Then$\xi$ = “number of heads” is$\text{Bin}(5,p)$ , and $$ P(\xi=2) = \binom{5}{2} p^2 (1-p)^3. $$
- Toss a coin
- Definition.
- A random variable
$\xi$ has a discrete uniform distribution on a finite set${s_1,\dots,s_n}$ if $$ p_\xi(x) = \frac{1}{n} I(x\in{s_1,\dots,s_n}), $$ written as$\xi\sim \text{Uni}(n)$ in the lecture.
- A random variable
- Intuition.
- Each of the
$n$ possible values is equally likely.
- Each of the
- Tiny example.
- Rolling a fair die:
$\xi$ = face value has $$ P(\xi = k) = 1/6,\quad k=1,\dots,6. $$
- Rolling a fair die:
- Definition.
- One version (used in the lecture):
$\xi\sim\text{Geo}(p)$ has pmf $$ p_\xi(x) = (1-p)^{x-1} p , I(x\in{1,2,3,\dots}). $$ - Alternative version sometimes used: $$ p_\xi(x) = (1-p)^x p , I(x\in{0,1,2,\dots}), $$ with slightly different interpretation.
- One version (used in the lecture):
- Intuition.
- First version: “trial number of the first success” in independent Bernoulli$(p)$ trials.
- Second version: “number of failures before the first success.”
- Tiny example.
- Toss a coin with success probability
$p$ until the first head. If$\xi$ is the trial number of the first head, then $$ P(\xi = x) = (1-p)^{x-1} p. $$
- Toss a coin with success probability
- Definition.
- A random variable
$\xi$ has Poisson distribution with parameter$\lambda>0$ , written$\xi\sim\text{Poi}(\lambda)$ , if $$ p_\xi(x) = e^{-\lambda} \frac{\lambda^x}{x!} I(x\in\mathbb{N}), $$ where$\mathbb{N} = {0,1,2,\dots}$ .
- A random variable
- Intuition.
- Models counts of rare, independent events over a fixed time or space interval (interpretation will be discussed in a later lecture).
- Tiny example (from lab).
- In a 20-second interval, the number of cars at an intersection may follow
$\text{Poi}(6)$ ; e.g., $$ P(X=2) = e^{-6}\frac{6^2}{2!}. $$
- In a 20-second interval, the number of cars at an intersection may follow
- Plain-language definition.
- A random vector is an ordered collection of random variables defined on the same probability space.
- The joint pmf describes probabilities of all combinations of their values; marginal pmfs give distributions of individual components; conditional pmfs describe one component given another.
- Formal definitions.
- Random vector
$X = (X_1,\dots,X_d): \Omega\to\mathbb{R}^d$ . - Joint pmf for discrete
$X$ : $$ p_X(x_1,\dots,x_d) = P(X_1=x_1,\dots,X_d=x_d), $$ with $$ \sum_{(x_1,\dots,x_d)} p_X(x_1,\dots,x_d) = 1. $$ - For
$d=2$ , marginals are $$ p_{X_1}(x) = \sum_y p_X(x,y),\quad p_{X_2}(y) = \sum_x p_X(x,y). $$ - For
$d=3$ , 2D marginals (projections) are sums over the remaining coordinate, e.g., $$ p_{X_1,X_2}(x,y) = \sum_z p_{X_1,X_2,X_3}(x,y,z), $$ and similarly for others. - Conditional pmf of
$X_1$ given$X_2=y$ (if$P(X_2=y)>0$ ): $$ p_{X_1|X_2}(x|y) = \frac{P(X_1=x,X_2=y)}{P(X_2=y)} = \frac{p_X(x,y)}{p_{X_2}(y)}. $$
- Random vector
- Intuition / mental model.
- Joint pmf captures full dependence between components; marginals forget some information; conditional pmfs tell you how one component behaves when the other is fixed.
- Independence of components shows up as factorization of the joint pmf:
$p_X(x,y) = p_{X_1}(x)p_{X_2}(y)$ .
- Tiny example (from lecture Example 3.1 idea).
- Two dice are thrown. Let
$X_1$ = sum of the two results,$X_2$ = difference. The pair$X=(X_1,X_2)$ is a random vector. Its joint pmf is defined on the set of possible$(\text{sum},\text{difference})$ pairs, and marginals can be obtained by summing over the other coordinate.
- Two dice are thrown. Let
- Formulas.
- Cdf:
$F_X(x)=P(X\le x)$ . - Pmf:
$p_X(x)=P(X=x)$ . - Relationship (discrete case with ordered support
$x_1<x_2<\dots$ ): $$ p_X(x_j) = F_X(x_j) - \lim_{x\to x_j^-} F_X(x). $$
- Cdf:
- When to use them.
- To compute probabilities like
$P(a\le X\le b)$ for discrete$X$ : sum$p_X(x)$ over integers in that range, or use differences of the cdf.
- To compute probabilities like
- Common mistakes.
- Mixing up cdf and pmf; the cdf is cumulative, the pmf is pointwise.
- Forgetting that the cdf for discrete
$X$ is right-continuous with jumps at points where$p_X(x)>0$ .
- Bernoulli$(p)$.
- $$ P(\xi=1)=p,\quad P(\xi=0)=1-p. $$
- Binomial$(n,p)$.
- $$ P(\xi=x) = \binom{n}{x} p^x(1-p)^{n-x},\quad x=0,1,\dots,n. $$
- Geometric$(p)$ (version counting trial of first success).
- $$ P(\xi=x) = (1-p)^{x-1}p,\quad x=1,2,\dots. $$
- Poisson$(\lambda)$.
- $$ P(\xi=x) = e^{-\lambda}\frac{\lambda^x}{x!},\quad x=0,1,2,\dots. $$
- Discrete uniform on
${s_1,\dots,s_n}$ .- $$ P(\xi = s_j) = 1/n,\quad j=1,\dots,n. $$
- When to use them.
- Bernoulli: single success/failure.
- Binomial: fixed number of independent trials with same success probability.
- Geometric: number of independent trials until first success.
- Poisson: counts of events in continuous time/space where events are rare and roughly independent (interpretation in later lectures).
- Common mistakes.
- Using Binomial when the number of trials is not fixed.
- Using Geometric when the probability of success changes between trials.
- Forgetting to restrict
$x$ to its proper support (e.g., nonnegative integers).
- Formula (discrete case).
- Random variables
$\xi_1,\dots,\xi_n$ are independent iff $$ P(\xi_1 = x_1,\dots,\xi_n=x_n) = \prod_{j=1}^n p_{\xi_j}(x_j) $$ for all$(x_1,\dots,x_n)$ .
- Random variables
- When to use it.
- To check whether two discrete random variables are independent: compute joint probabilities and compare to product of marginals.
- To construct a joint distribution from known marginals under an independence assumption.
- Common mistakes.
- Trying to verify independence using only marginal pmfs without checking joint probabilities.
- Assuming independence from just one equality like
$P(\xi_1=x_1|\xi_2=x_2)=P(\xi_1=x_1)$ for a single pair$(x_1,x_2)$ ; full independence must hold for all values.
- Formula.
- If
$X=(X_1,X_2)$ is a discrete random vector and$P(X_2=y)>0$ , then $$ p_{X_1|X_2}(x|y) = \frac{P(X_1=x,X_2=y)}{P(X_2=y)} = \frac{p_X(x,y)}{p_{X_2}(y)}. $$
- If
- When to use it.
- When you know or can compute the joint pmf and marginal pmf and need the conditional distribution of one component given another.
- To formalize statements like “given that the difference of the dice outcomes is 0, the sum is uniformly distributed over even numbers between 2 and 12.”
- Common mistakes.
- Forgetting to divide by
$P(X_2=y)$ . - Using the formula when
$P(X_2=y)=0$ ; in that case the conditional pmf is conventionally taken as 0 for all$x$ .
- Forgetting to divide by
- Setup.
- There are
$N$ people in a room. Assume 365 equally likely birthdays and ignore leap years. - Question: what is the probability that at least two people share a birthday?
- There are
- Step 1: define the sample space.
- Each person gets a birthday in
${1,\dots,365}$ . - A sample point is an ordered
$N$ -tuple$(\omega_1,\dots,\omega_N)$ with$\omega_j\in{1,\dots,365}$ . - The sample space
$$
\Omega = {(\omega_1,\dots,\omega_N): \omega_j\in{1,\dots,365}}
$$
has size
$365^N$ .
- Each person gets a birthday in
- Step 2: compute the probability that all birthdays are different.
- For all different birthdays, the first person has 365 choices, the second 364 choices, the third 363, and so on down to
$365-N+1$ . - Number of favorable outcomes:
$365\cdot 364\cdot \dots\cdot (365-N+1)$ . - Probability of all distinct birthdays: $$ P(\text{all different}) = \frac{365\cdot 364\cdot \dots\cdot (365-N+1)}{365^N}. $$
- For all different birthdays, the first person has 365 choices, the second 364 choices, the third 363, and so on down to
- Step 3: compute the probability of at least one match.
- Let
$A$ be the event “at least two people share a birthday”. Then$A^c$ is “all birthdays are different”. - So $$ P(A) = 1 - P(A^c) = 1 - \frac{365\cdot 364\cdot \dots\cdot (365-N+1)}{365^N}. $$
- The lecture notes that for
$N=23$ this probability already exceeds$1/2$ .
- Let
- Check your intuition.
- It is surprisingly likely to have a shared birthday with only 23 people; the key is that there are many possible pairs, not just matching your own birthday.
- Setup (from lecture).
- Toss a fair coin twice. Let
$X$ = number of heads in the two throws. - Sample space
$\Omega = {HH, HT, TH, TT}$ , each outcome has probability$1/4$ .
- Toss a fair coin twice. Let
- Step 1: compute the pmf.
-
$X(\text{TT}) = 0$ : one outcome, probability$1/4$ . -
$X(\text{HT}) = 1$ and$X(\text{TH}) = 1$ : two outcomes, combined probability$2/4=1/2$ . -
$X(\text{HH}) = 2$ : one outcome, probability$1/4$ . - So $$ p_X(0)=1/4,\quad p_X(1)=1/2,\quad p_X(2)=1/4,\quad p_X(x)=0 \text{ otherwise}. $$
-
- Step 2: compute the cdf
$F_X(x)=P(X\le x)$ .- For
$x<0$ , event${X\le x}$ is empty, so$F_X(x)=0$ . - For
$0\le x<1$ , only$X=0$ qualifies, so$F_X(x)=P(X=0)=1/4$ . - For
$1\le x<2$ ,$X=0$ or$X=1$ qualifies, so$F_X(x)=P(X=0)+P(X=1)=1/4+1/2=3/4$ . - For
$x\ge 2$ , all outcomes satisfy$X\le x$ , so$F_X(x)=1$ . - This matches the piecewise function given in the lecture.
- For
- Step 3: relate jumps of cdf to pmf.
- At
$x=0$ , jump size is$F_X(0)-\lim_{x\to0^-}F_X(x)=1/4-0=1/4=p_X(0)$ . - At
$x=1$ , jump size is$3/4-1/4=1/2=p_X(1)$ . - At
$x=2$ , jump size is$1-3/4=1/4=p_X(2)$ .
- At
- Check your intuition.
- This example shows how a discrete pmf and cdf encode the same information: the pmf gives probabilities at points, and the cdf is built from their cumulative sum.
- Joint distributions and indicator variables.
- Task 1: Let
$\xi$ be the number of heads in two fair coin tosses and$\eta = I(\xi=2)$ . Compute$P(\xi=x,\eta=y)$ for all suitable$(x,y)$ . - Task 2: Let
$\xi$ be Bernoulli$(p)$, define$\eta = 1-\xi$ ,$\zeta = \xi\eta$ . Compute joint distributions$P(\xi=x,\eta=y)$ and$P(\xi=x,\zeta=z)$ for$0\le x,y,z\le1$ .
- Task 1: Let
- Derived discrete distributions and pmfs.
- Task 3: Masha tosses
$n$ coins, each showing tails with probability$p$ . Tails are tossed again once. Find the pmf of the number of tails in the second round. - Task 4: Independent random variables
$\xi$ and$\eta$ with the same pmf$p(x)=2^{-x}I(x\in\mathbb{N})$ . Find$P(\min{\xi,\eta}\le x)$ ,$P(\eta>\xi)$ , and$P(\xi\text{ divides }\eta)$ .
- Task 3: Masha tosses
- Sums of discrete random variables and conditional distributions.
- Task 5: Independent Poisson r.v.s
$\xi\sim\text{Poi}(\mu)$ ,$\eta\sim\text{Poi}(\nu)$ . Show that$\xi+\eta\sim\text{Poi}(\mu+\nu)$ and that the conditional distribution of$\xi$ given$\xi+\eta=n$ is binomial; find its parameters. - Task 6: Prove that the sum of independent
$\text{Bin}(n,p)$ and$\text{Bin}(m,p)$ is$\text{Bin}(n+m,p)$ .
- Task 5: Independent Poisson r.v.s
- Additional self-practice problems.
- Checking that a given function is a valid pmf by finding a normalizing constant
$c$ . - Drawing cdfs for simple discrete distributions.
- Computing probabilities for Poisson counts (e.g., cars at an intersection).
- Modeling system reliability with binomial distributions (operational components).
- Checking that a given function is a valid pmf by finding a normalizing constant
- Joint distributions involving indicators and Bernoulli variables.
- Identify which combinations of values are possible; others automatically have probability 0.
- Use the definition of indicator:
$\eta=I(\xi=2)$ means$\eta=1$ only when$\xi=2$ , and$\eta=0$ otherwise. - For
$\xi,\eta,\zeta$ built from a Bernoulli variable, write out the two possible values of$\xi$ (0 or 1) and compute$\eta$ and$\zeta$ deterministically from them, then assign probabilities.
- Constructing pmfs from two-stage experiments (Masha’s coins).
- Let the number of tails in the first round be a Binomial r.v. (with parameter
$p$ and$n$ trials). - The second round involves tossing only those coins that were tails; the number of second-round tails is then Binomial with random number of trials.
- Use the law of total probability over possible first-round counts and sum appropriately.
- Let the number of tails in the first round be a Binomial r.v. (with parameter
- Working with infinite discrete distributions.
- For
$\xi,\eta$ with pmf$p(x)=2^{-x}$ :- To find
$P(\min{\xi,\eta}\le x)$ , think of the complement$P(\min{\xi,\eta}>x) = P(\xi>x,\eta>x)$ and use independence. - For
$P(\eta>\xi)$ , use symmetry or explicit summation over pairs$(k,\ell)$ with$\ell>k$ . - For
$P(\xi\text{ divides }\eta)$ , sum over all integer pairs where$\eta$ is a multiple of$\xi$ .
- To find
- For
- Poisson and binomial sums and conditionals.
- For the Poisson sum
$\xi+\eta$ : use the joint pmf of independent Poisson variables and sum over all ways to split total count$n$ into$k$ and$n-k$ . - Show that the resulting pmf matches a Poisson$(\mu+\nu)$ form.
- For the conditional distribution
$P(\xi=k|\xi+\eta=n)$ , apply the definition of conditional probability with the joint pmf and simplify to recognize a binomial pmf in$k$ . - For the binomial sum, work with pmfs or use the interpretation as total successes in
$n+m$ independent trials.
- For the Poisson sum
- Valid pmfs and cdfs.
- When a function
$f(x)$ is proposed as a pmf on a finite set of$x$ -values, ensure$f(x)\ge 0$ and solve for the constant$c$ in$\sum f(x)=1$ . - To draw the cdf from a table of values: order the support, then cumulatively add probabilities and plot the resulting step function.
- When a function
- Practice 1: identifying a Bernoulli variable from an indicator.
- Question: Let
$E$ be the event “the number of heads in two tosses is 2”, and let$\eta = I(E)$ . What is the distribution of$\eta$ ? - Brief answer:
$\eta$ takes value 1 with probability$P(E)=1/4$ and 0 with probability$3/4$ , so$\eta\sim\text{Bernoulli}(p)$ with$p=1/4$ .
- Question: Let
- Practice 2: recognizing a binomial distribution.
- Question: A system has 5 independent components, each working with probability 0.92. Let
$X$ be the number of working components. What is the distribution and pmf of$X$ ? - Brief answer:
$X\sim\text{Bin}(5,0.92)$ with $$ P(X=x) = \binom{5}{x} 0.92^x (0.08)^{5-x},\quad x=0,1,2,3,4,5. $$
- Question: A system has 5 independent components, each working with probability 0.92. Let
- Practice 3: sum of independent Poisson variables.
- Question:
$\xi\sim\text{Poi}(2)$ and$\eta\sim\text{Poi}(3)$ are independent. What is the distribution of$\xi+\eta$ ? - Brief answer: By the lab result,
$\xi+\eta\sim\text{Poi}(5)$ .
- Question:
- Random variables are measurable functions from the sample space to the real numbers; indicator variables connect events and random variables.
- Every random variable has a distribution function (cdf)
$F_X(x)=P(X\le x)$ that is non-decreasing, right-continuous, and goes from 0 to 1. - Discrete random variables “live” on a countable set; their pmf
$p_X(x)=P(X=x)$ sums to 1 and determines the cdf via jumps. - Independence is extended from events to random variables; for discrete variables, independence means the joint pmf factors as the product of the marginals.
- Standard discrete distributions introduced: Bernoulli, Binomial, discrete uniform, Geometric, and Poisson, each modeling a different type of counting process.
- Random vectors group several random variables; their behavior is described using joint, marginal, and conditional pmfs.
- The lab reinforces these ideas through problems on joint distributions, indicator variables, derived pmfs, sums of Poisson and binomial variables, and constructing valid discrete distributions.