auto-update(nvim): 2025-02-19 16:36:00
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@ -242,10 +242,7 @@ Requires equally likely outcomes and finite sample spaces.
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An approach done commonly by applied statisticians who work in the disgusting
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real world. This is where we are generally concerned with irrelevant concerns
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like accurate sampling and $p$-values and such. I am told this is covered in
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PSTAT 120B, so hopefully I can avoid ever taking that class (as a pure math
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major).
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like accurate sampling and $p$-values and such.
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$
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P(A) = (hash "of times" A "occurs in large number of trials") / (hash "of trials")
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$
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@ -768,6 +765,72 @@ us generalize to more than two colors.
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Both approaches given the same answer.
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]
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= Baye's theorem and conditional probability
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== Conditional probability, partitions, law of total probability
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Sometimes we want to analyze the probability of events in a sample space given
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that we already know another event has occurred. Ergo, we want the probability
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of $A in Omega$ conditional on the event $B in Omega$.
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#definition[
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For two events $A, B in Omega$, the probability of $A$ given $B$ is written
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$
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P(A | B)
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$
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]
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#fact[
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To calculate the conditional probability, use the following formula:
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$
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P(A | B) = (P(A B)) / (P(B))
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$
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]
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Oftentimes we don't know $P(B)$, but we do know $P(B)$ given some events in
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$Omega$. That is, we know the probability of $B$ conditional on some events.
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For example, if we have a 50% chance of choosing a rigged (6-sided) die and a
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50% chance of choosing a fair die, we know the probability of getting side $n$
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given that we have the rigged die, and the probability of side $n$ given that
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we have the fair die. Also note that we know the probability of both events
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we're conditioning on (50% each), and they're disjoint events.
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In these situations, the following law is useful:
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#theorem[Law of total probability][
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Given a _partition_ of $Omega$ with pairwise disjoint subsets $A_1, A_2, A_3, ..., A_n in Omega$, such that
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$
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union.big_(A_i in Omega) A_i = Omega \
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sect.big_(A_i in Omega) A_i = emptyset
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$
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The probability of an event $B in Omega$ is given by
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$
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P(B) = P(B | A_1) P(A_1) + P(B | A_2) P(A_2) + dots.c + P(B | A_n) P(A_n)
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$
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]<law-total-prob>
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#proof[
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This is easy to show by writing the definition of the conditional probability
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and simplifying.
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]
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== Baye's theorem
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Finally let's discuss a rule for inverting conditional probabilities, that is,
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getting $P(B | A)$ from $P(A | B)$.
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#theorem[Baye's theorem][
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Given two events $A,B in Omega$,
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$
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P(A | B) = (P(B | A)P(A)) / (P(B | A)P(A) + P(B | A^c)P(A^c))
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$
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]
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#proof[
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Apply the definition of conditional probability, then apply @law-total-prob
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noting that $A$ and $A^c$ are a partitioning of $Omega$.
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]
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= Lecture #datetime(day: 23, month: 1, year: 2025).display()
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== Independence
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@ -777,7 +840,9 @@ us generalize to more than two colors.
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"Joint probability is equal to product of their marginal probabilities."
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]
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#fact[This definition must be used to show the independence of two events.]
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#fact[
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This definition must be used to show the independence of two events.
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]
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#fact[
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If $A$ and $B$ are independent, then,
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@ -836,6 +901,99 @@ us generalize to more than two colors.
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different events $A_i$ and $A_j$ are independent for any $i != j$.
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]
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= A bit of review on random variables
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== Random variables, discrete random variables
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Recall that a random variable $X$ is a function $X : Omega -> RR$ that gives
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the probability of an event $omega in Omega$. The _probability distribution_ of
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$X$ gives its important probabilistic information. The probability distribution
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is a description of the probabilities $P(X in B)$ for subsets $B in RR$. We
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describe the probability density function and the cumulative distribution
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function.
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A random variable $X$ is discrete if there is countable $A$ such that $P(X in
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A) = 1$. $k$ is a possible value if $P(X = k) > 0$.
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A discrete random variable has probability distribution entirely determined by
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p.m.f $p(k) = P(X = k)$. The p.m.f. is a function from the set of possible
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values of $X$ into $[0,1]$. Labeling the p.m.f. with the random variable is
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done by $p_X (k)$.
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By the axioms of probability,
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$
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sum_k p_X (k) = sum_k P(X=k) = 1
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$
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For a subset $B subset RR$,
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$
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P(X in B) = sum_(k in B) p_X (k)
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$
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== Continuous random variables
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Now we introduce another major class of random variables.
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#definition[
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Let $X$ be a random variable. If $f$ satisfies
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$
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P(X <= b) = integral^b_(-infinity) f(x) dif x
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$
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for all $b in RR$, then $f$ is the *probability density function* of $X$.
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]
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The probability that $X in (-infinity, b]$ is equal to the area under the graph
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of $f$ from $-infinity$ to $b$.
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A corollary is the following.
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#fact[
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$ P(X in B) = integral_B f(x) dif x $
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]
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for any $B subset RR$ where integration makes sense.
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The set can be bounded or unbounded, or any collection of intervals.
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#fact[
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$ P(a <= X <= b) = integral_a^b f(x) dif x $
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$ P(X > a) = integral_a^infinity f(x) dif x $
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]
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#fact[
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If a random variable $X$ has density function $f$ then individual point
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values have probability zero:
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$ P(X = c) = integral_c^c f(x) dif x = 0, forall c in RR $
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]
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#remark[
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It follows a random variable with a density function is not discrete. Also
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the probabilities of intervals are not changed by including or excluding
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endpoints.
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]
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How to determine which functions are p.d.f.s? Since $P(-infinity < X <
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infinity) = 1$, a p.d.f. $f$ must satisfy
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$
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f(x) >= 0 forall x in RR \
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integral^infinity_(-infinity) f(x) dif x = 1
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$
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#fact[
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Random variables with density functions are called _continuous_ random
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variables. This does not imply that the random variable is a continuous
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function on $Omega$ but it is standard terminology.
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]
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Named distributions of continuous random variables are introduced in the
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following chapters.
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= Lecture #datetime(day: 27, year: 2025, month: 1).display()
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== Bernoulli trials
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@ -1138,107 +1296,7 @@ exactly one sequence that gives us success.
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$
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]
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= Notes on textbook chapter 3
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Recall that a random variable $X$ is a function $X : Omega -> RR$ that gives
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the probability of an event $omega in Omega$. The _probability distribution_ of
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$X$ gives its important probabilistic information. The probability distribution
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is a description of the probabilities $P(X in B)$ for subsets $B in RR$. We
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describe the probability density function and the cumulative distribution
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function.
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A random variable $X$ is discrete if there is countable $A$ such that $P(X in
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A) = 1$. $k$ is a possible value if $P(X = k) > 0$.
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A discrete random variable has probability distribution entirely determined by
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p.m.f $p(k) = P(X = k)$. The p.m.f. is a function from the set of possible
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values of $X$ into $[0,1]$. Labeling the p.m.f. with the random variable is
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done by $p_X (k)$.
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By the axioms of probability,
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$
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sum_k p_X (k) = sum_k P(X=k) = 1
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$
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For a subset $B subset RR$,
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$
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P(X in B) = sum_(k in B) p_X (k)
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$
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Now we introduce another major class of random variables.
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#definition[
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Let $X$ be a random variable. If $f$ satisfies
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$
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P(X <= b) = integral^b_(-infinity) f(x) dif x
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$
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for all $b in RR$, then $f$ is the *probability density function* of $X$.
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]
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The probability that $X in (-infinity, b]$ is equal to the area under the graph
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of $f$ from $-infinity$ to $b$.
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A corollary is the following.
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#fact[
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$ P(X in B) = integral_B f(x) dif x $
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]
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for any $B subset RR$ where integration makes sense.
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The set can be bounded or unbounded, or any collection of intervals.
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#fact[
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$ P(a <= X <= b) = integral_a^b f(x) dif x $
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$ P(X > a) = integral_a^infinity f(x) dif x $
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]
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#fact[
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If a random variable $X$ has density function $f$ then individual point
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values have probability zero:
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$ P(X = c) = integral_c^c f(x) dif x = 0, forall c in RR $
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]
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#remark[
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It follows a random variable with a density function is not discrete. Also
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the probabilities of intervals are not changed by including or excluding
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endpoints.
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]
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How to determine which functions are p.d.f.s? Since $P(-infinity < X <
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infinity) = 1$, a p.d.f. $f$ must satisfy
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$
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f(x) >= 0 forall x in RR \
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integral^infinity_(-infinity) f(x) dif x = 1
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$
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#fact[
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Random variables with density functions are called _continuous_ random
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variables. This does not imply that the random variable is a continuous
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function on $Omega$ but it is standard terminology.
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]
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#definition[
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Let $[a,b]$ be a bounded interval on the real line. A random variable $X$ has
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the *uniform distribution* on $[a,b]$ if $X$ has density function
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$
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f(x) = cases(
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1/(b-a)", if" x in [a,b],
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0", if" x in.not [a,b]
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)
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$
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Abbreviate this by $X ~ "Unif"[a,b]$.
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]
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= Notes on week 3 lecture slides
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= Some more discrete distributions
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== Negative binomial
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