Thursday, Jan 22

Section Thursday, Jan 22

This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.

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Subsection Discrete Distributions

Theorem 40.

Suppose we expand \((x + y)^n\text{.}\) We’ll get a polynomial with some coefficients:

\begin{gather*} (x+y)^n = {n\choose 0} x^n + {n\choose 1} x^{n-1}y + \dotsb + {n\choose n-1} xy^{n-1} + {n\choose n} y^n, \end{gather*}

where \({n \choose k}\text{,}\) read "\(n\) choose \(k\)", is the coefficient of \(x^{n-k}y^k\text{.}\)

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Alternatively: \({n\choose k}\) is the number of ways to pick \(k\) things out of a set of \(n\) things. Then:

\begin{gather*} {n \choose k} = \frac{n!}{k!(n-k)!}, \end{gather*}

where \(n!\text{,}\) read "\(n\) factorial", refers to the product:

\begin{gather*} n! = n(n-1)(n-2)\dotsm (3)(2)(1). \end{gather*}

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Fact 41.

\(0! = 1.\)

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Example 42.

\begin{align*} {n\choose 0} \amp = \frac{n!}{0!(n - 0)!} = \frac{n!}{0!}{n!} = \frac{1}{0!} = 1 \\ {10 \choose 3} \amp = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8 \times 7!}{3\times 2 \times 1 \times 7!} = 10 \times 3 \times 4 = 120. \end{align*}

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So now we get a general probability formula for the binomial distribution:

\begin{gather*} b(k; n, p) = {n \choose k} p^k (1-p)^{n-k}. \end{gather*}

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Let’s see some more important distribution types:

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Example 43.

Suppose we have a coin with bias \(p\text{,}\) i.e., probability \(p\) of coming up heads. (Note: this is not a common term, but it will be convenient for us to have some terminology for it since we’ll refer to this parameter often.) We flip the coin repeatedly until we see heads. Let \(N\) be the number of flips.

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Definition 44.

\(N\) has the geometric distribution with parameter \(p\text{.}\) We’ll write \(N \sim \Geom(p)\text{,}\) and we’ll write:

\begin{gather*} g(k) = g(k; p) = \Pr(N = k). \end{gather*}

The geometric distribution is given by:

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Table 45. Geometrid Distribution

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\(k\)	flip sequences	\(g(k)\)
1	H	\(p\)
2	TH	\((1-p)p\)
3	TTH	\((1-p)^2p\)
4	TTTH	\((1-p)^3p\)

In general:

\begin{gather*} g(k; p) = (1-p)^{k-1}p. \end{gather*}

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Example 46.

Suppose a toxin molecule inside a cell has a 0.2 chance of leaving during each minute. Let \(T\) be the number of minutes until the molecule leaves. Find \(\Pr(T \leq 3).\)

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If \(T \leq 3\text{,}\) then \(T\) is either 1, 2, or 3.

\begin{align*} \Pr(T = 1) \amp = g(1; 0.2) = 0.2 \\ \Pr(T = 2) \amp = g(2; 0.2) = 0.8 \times 0.2 = 0.16 \\ \Pr(T = 3) \amp = g(3; 0.2) = 0.8^2 \times 0.2 = 0.128 \\ \Pr(T \leq 3) \amp = 0.2 + 0.16 + 0.128 = 0.488. \end{align*}

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Definition 47.

A Poisson process is a process in which some event occurs at a constant probabilistic rate \(\lambda\text{.}\) Suppose we observe a Poisson process for \(t\) time. Let \(N\) be the number of occurrences of the event during that observation time. Then \(N\) has the Poisson distribution with parameters \(\lambda, t\text{.}\) We’ll write \(N \times \Poiss(\lambda, t)\text{,}\) and:

\begin{gather*} p(k) = p(k; \lambda, t) = \Pr(N = k) = \frac{(\lambda t)^k}{k!} e^{-\lambda t}. \end{gather*}

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Example 48.

Consider the following examples:

A radioactive material emmitting particles as it decays.
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A call center receiving calls.
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A stretch of highways seeing cars pass by.
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Example 49.

Suppose a highway typically has 200 cars per hour. Let \(N\) be the number of cars in a 30-minute observation period. Then \(\lambda = 200, t = \frac{1}{2}\text{,}\) so \(\lambda t = 100\) . Then:

\begin{align*} \Pr(N = k) \amp = \frac{100^k}{k!} e^{-100} \\ \text{e.g., } \Pr(N = 110) \amp = \frac{100^{110}}{110!} e^{-100} \approx 0.023. \end{align*}

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Subsection Continuous Distributions

Example 50.

Suppose we pick a real number \(X \in [0, 4]\) uniformly. Intuitively, we can say things like \(\Pr(X \leq 1) = \frac{1}{4}\) and \(\Pr(1 \leq X \leq 3) = \frac{1}{2}\text{.}\) What aobut \(\Pr(X = 2)?\)

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Assigning probabilities to individual outcomes isn’t useful here. Instead, we assign probabilities to intervals of the form \(a \leq X \leq b\text{.}\)

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Definition 51.

Let \(X\) be a random variable. If \(X\) takes an interval’s worth of values, then it’s called continuous. Otherwise, it’s called discrete.

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Example 52.

Indicator, Bin, Geom, Poiss are all discrete.
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Pick \(X \in [0, 4]\) is continuous.
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Height of a plant is continuous.
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Time is often (but not always) treated as continuous.
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Definition 53.

Let \(X\) be a continuous random variable. A probability density function (pdf) for \(X\) is a function \(f(x)\) such that:

\(f(x) \geq 0\) for all \(x\text{,}\) and,
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\(\int_{\Omega} f(x)\ dx = 1\text{,}\) where \(\int_{\Omega}\) indicates that we should integrate over the entire range of possible values of \(X\text{.}\)
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Given a pdf \(f(x)\) for \(X\text{:}\)

\begin{gather*} \Pr(a \leq X \leq b) = \int_a^b f(x)\ dx. \end{gather*}

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Example 54.

Pick \(X\in [0, 4]\) uniformly. Because of the uniform assumption, \(f(x)\) must be a constant function \(f(x) = k\) for some \(k\text{.}\) To find \(k\text{:}\)

\begin{align*} 1 = \int_0^4 f(x)\ dx \amp = \int_0^4 k\ dx = kx \bigg|_0^4 = 4k - 0 \\ \Rightarrow \quad k \amp = \frac{1}{4} \end{align*}

More generally, if \(X \in [a, b]\) is chosen uniformly, then the pdf would be:

\begin{gather*} f(x) = \frac{1}{b - a}, \quad a \leq x \leq b \end{gather*}

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