class: center, middle, inverse layout: yes name: inverse ## STAT 305: Chapter 5 ### Part III ### Amin Shirazi .footnote[Course page: [ashirazist.github.io/stat305_s2020.github.io](https://ashirazist.github.io/stat305_s2020.github.io/)] --- layout: true class: center, middle, inverse --- # Continuous Random Variables ## Terminology, Use, and Common Distributions --- # What is a Continuous Random Variable? --- layout:false .left-column[ ## Background ### What? ] .right-column[ ## Background on Continuous Random Variable Along with discrete random variables, we have continuous random variables. While discrete random variables take one specific values from a _discrete_ (aka countable) set of possible real-number values, continous random variables take values over intervals of real numbers. >**def: Continuous random variable ** </br> >A continuous random variable is a random variable which takes values on a continuous interval of real numbers. The reason we treat them differently has mainly to do with the differences in how the math behaves: now that we are dealing with interval ranges, we change summations to integrals. ] --- layout:false .left-column[ ## Background ### What? ] .right-column[ ###Examples of continuous random variable: > **Z** is the amount of torque required to lossen the next bold (not rounded) > **T** is the time you will wait for the next bus > **C** is the outside temprature at 11:49 pm tomorrow > **L** is the length of the next manufactured metal bar > **V** is the `\(%\)` yield of the next run of process ] --- layout: true class: center, middle, inverse --- # Terminology and Usage --- layout:false .left-column[ ## Background ## Terminology ### pdf ] .right-column[ ### Probability Density Function Since we are now taking values over an interval, we can not "add up" probabilities with our probability function anymore. Instead, we need a new function to describe probability: >**def: probability density function** </br> >A probability density function (pdf) defines the way the probability of a continuous random variable is distributed across the interval of values it can take. Since it represents probability, the probability function must always be non-negative. Regions of higher density have higher probability. ] --- layout:false .left-column[ ## Background ## Terminology ### pdf ] .right-column[ ### Probability Density Function ####Validity of a *pdf* Any function that satisfies the following can be a probability density function: 1. `\(\int_{-\infty}^{\infty} f(x) dx = 1\)` 2. `\(f(x) \ge 0\)` for all `\(x\)` in `\((-\infty, \infty)\)` and such that for all `\(a \le b\)`, `$$P(a \le X \le b) = P(a \le X < b) =\\ P(a < X \le b) = P(a < X < b)\\ =\int\limits_a^bf(x)dx.$$` ] --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ] .right-column[ ### Probability Density Function With continuous random variables, we use pdfs to get probabilities as follows: >For a continuous random variable `\(X\)` with probability density function `\(f(x)\)`, >$$P(a \le X \le b) = \int_{a}^{b} f(x) dx$$ >for any real values `\(a, b\)` such that `\(a\le b\)` <!-- --> ] --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ] .right-column[ ###Example Consider a de-magnetized compass needle mounted at its center so that it can spin freely. It is spun clockwise and when it comes to rest the angle, `\(\theta\)`, from the vertical, is measured. Let `$$Y = \text{the angle measured after each spin in radians}$$` What values can `\(Y\)` take? &nbsp; What form makes sense for `\(f(y)\)`? ] --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ] .right-column[ ###Example If this form is adopted, that what must the pdf be? &nbsp; &nbsp; &nbsp; &nbsp; Using this pdf, calculate the following probabilities: - `\(P[Y < \frac{\pi}{2}]\)` ] --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ] .right-column[ ###Example - `\(P[\frac{\pi}{2} < Y < 2\pi]\)` &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; - `\(P[Y = \frac{\pi}{6}]\)` ] --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ### cdf ] .right-column[ ### Cumulative Density Function (**CDF**) We also have the cumulative density function for continuous random variables: >**def: Cumulative density function (cdf)** >For a continous random variable, `\(X\)`, with pdf f(x) the cumulative density function `\(F(x)\)` is defined as the probability that `\(X\)` takes a value less than or equal to `\(x\)` which is to say >$$ F(x) = P(X \le x) = \int_{-\infty}^{x} f(t) dt $$ TRUE FACT: the Fundamental Theorem of Calculus applies here: $$ \dfrac{d}{dx} F(x) = f(x) $$ ] --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ### cdf ] .right-column[ ### Cumulative Density Function (**CDF**) #### Properties of CDF for continuous random variables As with discrete random variables, `\(F\)` has the following properties: - **F** is monotonically increasing (i.e it is never decreasing) - `\(\lim_{x\rightarrow-\infty}{F(x)}= 0\)` and `\(\lim_{x\rightarrow+\infty}{F(x)}= 1\)` - This means that `\(0\leq{F(x)}\le 1\)` for **any CDF** - **F** is *continuous*. (instead of just right continuous in discrete form) ] --- layout: true class: center, middle, inverse --- ##Mean and Variance ###of ##Continuous Random Variables --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ### cdf ### E(X), V(X) ] .right-column[ ### Expected Value and Variance #### Expected Value As with discrete random variables, continuous random variables have expected values and variances: >**def: Expected Value of Continuous Random Variable** </br> >For a continous random variable, `\(X\)`, with pdf f(x) the expected value (also known as the mean) is defined as >$$ E(X) = \int_{-\infty}^{\infty} x f(x) dx $$ We often use the symbol `\(\mu\)` for the mean of a random variable, since writing `\(E(X)\)` can get confusing when lots of other parenthesis are around. We also sometimes write `\(EX\)`. ] --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ### cdf ### E(X), V(X) ] .right-column[ ### Expected Value and Variance #### Variance >**def: Variance of Continuous Random Variable** </br> >For a continous random variable, `\(X\)`, with pdf f(x) and expected value `\(\mu\)`, the variance is defined as >$$ V(X) = \int_{-\infty}^{\infty} (x - \mu)^2 \cdot f(x) dx $$ >which is identical to saying >$$ V(X) = E(X^2) - E(X)^2 $$ We will sometimes use the symbol `\(\sigma^2\)` to refer to the variance and you may see the notation `\(Var X\)` or `\(VX\)` as well. ] --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ### cdf ### E(X), V(X) ] .right-column[ ### Expected Value and Variance #### Sdandard Deviation (SD) We can also use the variance to get the standard deviation of the random variable: >**def: Standard Deviation of Continuous Random Variable** </br> >For a continous random variable, `\(X\)`, with pdf f(x) and expected value `\(\mu\)`, the standard deviation is defined as: >$$ \sigma = \sqrt{\sigma^2} = \sqrt{\int_{-\infty}^{\infty} (x - \mu)^2 \cdot f(x) dx} $$ ] --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ] .right-column[ ### Expected Value and Variance: Example ####Library books Let `\(X\)` denote the amount of time for which a book on `\(2\)`-hour hold reserve at a college library is checked out by a randomly selected student and suppose its density function is $$ f(x) = \begin{cases} 0.5x & 0 \le x \le 2 \\\\ 0 & \text{otherwise} \end{cases} $$ Calculate `\(\text{E}X\)` and `\(\text{Var}X\)`. ] --- layout: true class: center, middle, inverse --- ##An important point about Expected Value ##and Variance of Random Variables --- layout:false .left-column[ ## Background ## Terms and Use ### pdf ] .right-column[ ### Expected Value and Variance: For a linear function, `\(g(X) = aX + b\)`, where `\(a\)` and `\(b\)` are constants, > `\(\text{E}(aX + b)= a \text{E}(X) + b\)` > `\(\text{Var}(aX + b)= a^2 \text{Var}(X)\)` e.g Let `\(X\sim Binomial(5, 0.2)\)`. What is the expected value and variance of 4X- 3? ] --- layout: true class: center, middle, inverse --- # Common Distributions ## Uniform Distribution --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ] .right-column[ ## Common continuous Distributions ### Uniform Distribution For cases where we only know/believe/assume that a value will be between two numbers but know/believe/assume _nothing_ else. **Origin**: We know a the random variable will take a value inside a certain range, but we don't have any belief that one part of that range is more likely than another part of that range. >**Definition: Uniform random variable **</br> >The random variable `\(U\)` is a uniform random variable on the interval `\([a, b]\)` if it's density is constant on `\([a, b]\)` and the probability it takes a value outside `\([a, b]\)` is 0. We say that `\(U\)` follows a uniform distribution or `\(U \sim uniform(a, b)\)`. ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ] .right-column[ ### Uniform Distribution >**Definition: Uniform pdf** </br> >If `\(U\)` is a uniform random variable on `\([a, b]\)` then the probability density function of `\(U\)` is given by >$$f(u) = \begin{cases} > \dfrac{1}{b-a} & a \le u \le b \\\\ > 0 & o.w. > \end{cases} >$$` With this, we can find the for any value of `\(a\)` and `\(b\)`, if `\(U \sim uniform(a, b)\)` the mean and variance are: $$ E(U) = \frac{1}{2}(b-a) $$ $$ Var(U) = \frac{1}{12}(b-a)^2 $$ ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ] .right-column[ ### Uniform Distribution >**Definition: Uniform cdf** </br> >If `\(U\)` is a uniform random variable on `\([a, b]\)` then the cumulative density function of `\(U\)` is given by >$$F(u) = \begin{cases} > 0 & u < a \\\\ > \dfrac{u-a}{b-a} & a \le u \le b \\\\ > 1 & u > b \\\\ >\end{cases} >$$` ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ] .right-column[ ### Uniform Distribution A few useful notes: - The most commonly used uniform random variable is `\(U \sim Uniform(0,1)\)`. - Again, this is useful if we want to use a random variable that takes values within an interval, but we don't think it is likely to be in any certain region. - The values `\(a\)` and `\(b\)` used to determine the range in which `\(f(u)\)` is not 0 are parameters of the distribution. ] --- layout:true class: middle, center, inverse --- # Common Continuous Distributions ##Exponential Distribution --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ] .right-column[ ### Exponential Distribution **Definition: Exponential random variable ** An `\(\text{Exp}(\alpha)\)` random variable measures the waiting time until a specific event that has an equal chance of happening at any point in time. (it can be cosidered the continous version of geometric distribution) >Examples: >- Time between your arrival at the bus station and the moment that bus arrives > >&nbsp; > >- Time until the next person walks inside the park's library > >&nbsp; > >- The time (in hours) until a light bulb burns out. ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ] .right-column[ ### Exponential Distribution >**Definition: Exponential pdf** </br> >If `\(X\)` is an exponential random variable with rate `\(\frac{1}{\alpha}\)` then the probability density function of `\(X\)` is given by >$$f(u) = \begin{cases} >\dfrac{1}{\alpha} e^{-\frac{x}{\alpha}} & x \ge 0 \\\\ >0 & o.w. >\end{cases}$$` <!-- --> ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ] .right-column[ ### Exponential Distribution >**Definition: Exponential CDF** </br> >If `\(X\)` is a exponential random variable with rate `\(1/\alpha\)` then the cumulative density function of `\(X\)` is given by >$$F(x) = \begin{cases} >1 - exp(-x/\alpha) & 0 \le x \\\\ >0 & x < 0 \\\\ >\end{cases} >$$` <!-- --> ] --- layout:true class: middle, center, inverse --- ##Mean and Variance of ##Exponential Distribution --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ] .right-column[ ### Exponential Distribution >**Definition: Exponential pdf** </br> >If `\(X\)` is an exponential random variable with rate `\(\frac{1}{\alpha}\)` then the probability density function of `\(X\)` is given by >$$f(u) = \begin{cases} >\dfrac{1}{\alpha} e^{-\frac{x}{\alpha}} & x \ge 0 \\\\ >0 & o.w. >\end{cases} >$$` From this, we can derive: $$ E(X) = \alpha $$ $$ Var(X) = \alpha^2 $$ ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ] .right-column[ ### Exponential Distribution **Example**: Library arrivals, cont'd Recall the example the arrival rate of students at Parks library between 12:00 and 12:10pm early in the week to be about `\(12.5\)` students per minute. That translates to a `\(1/12.5 = .08\)` minute average waiting time between student arrivals. Consider observing the entrance to Parks library at exactly noon next Tuesday and define the random variable $$ `\begin{align} T&: \text{the waiting time (min) until the first} \\ & \text{student passes through the door.}\\ \end{align}` $$ Using `\(T \sim \text{Exp}(.08)\)`, what is the probability of waiting more than `\(10\)` seconds (1/6 min) for the first arrival? ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ] .right-column[ ### Exponential Distribution **Example**: Library arrivals, cont'd $$ `\begin{align} T&: \text{the waiting time (min) until the first} \\ & \text{student passes through the door.}\\ \end{align}` $$ What is the probability of waiting less than `\(5\)` seconds? ] --- layout: true class: center, middle, inverse --- ##Common Continous Distibutions ##Normal Distribution --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ] .right-column[ ### The Normal distribution We have already seen the normal distribution as a "bell shaped" distribution, but we can formalize this. The **normal** or **Gaussian** `\((\mu, \sigma^2)\)` distribution is a continuous probability distribution with probability density function (pdf) `$$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x - \mu)^2/{2\sigma^2}} \qquad \text{for all } x$$` for `\(\sigma > 0\)`. We then show that by `\(X\sim\text{N}(\mu, \sigma^2)\)` ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ] .right-column[ ### The Normal distribution A normal random variable is (often) a finite average of many repeated, independent, identical trials. >Mean width of the next 50 hexamine pallets > >Mean height of 30 students > >Total `\(\%\)` yield of the next 10 runs of a chemical process ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ] .right-column[ ### Normal Distribution's Center and Shape Regardless of the values of `\(\mu\)` and `\(\sigma^2\)`, the normal pdf has the following shape: <center> <img src="normal_pdf.png" alt="normal_pdf" width="400"/> </center> In other words, the distribution is centered around `\(\mu\)` and has an inflection point at `\(\sigma = \sqrt{\sigma^2}\)`. In this way, the value of `\(\mu\)` determines the center of our distribution and the value of `\(\sigma^2\)` deterimes the spread. ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ] .right-column[ ### Normal Distribution's Center and Shape Here we can see what differences in `\(\mu\)` and `\(\sigma^2\)` do to the shape of the shape of distribution <center> <img src="normal_comparisons.png" alt="normal_comparisons" width="600" height= "400"/> </center> ] --- layout: true class: center, middle, inverse --- ##Mean and Variance ###of ##Normal Distribution --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ] .right-column[ ### The Normal distribution It is not obvious, but - `\(\int\limits_{-\infty}^\infty f(x) dx = \int\limits_{-\infty}^\infty \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x - \mu)^2/{2\sigma^2}} dx =\)` &nbsp; - `\(\text{E}X = \int\limits_{-\infty}^\infty x \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x - \mu)^2/{2\sigma^2}} dx =\)` &nbsp; - `\(\text{Var}X = \int\limits_{-\infty}^\infty (x - \mu)^2 \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x - \mu)^2/{2\sigma^2}} dx =\)` ] --- layout: true class: center, middle, inverse --- ##One poine before we go on ###*Standardization* --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ] .right-column[ ###Definition **Standardization** is the process of transforming a random variable, `\(X\)`, into the signed number of standard deviations by which it is is above its mean value. $$ Z = \frac{X - \text{E}X}{\text{SD}(X)} $$ `\(Z\)` has mean `\(0\)` &nbsp; &nbsp; &nbsp; `\(Z\)` has variance (and standard deviation) `\(1\)` ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ] .right-column[ &nbsp; &nbsp; The Calculus I methods of evaluating integrals via anti-differentiation will fail when it comes to normal densities. They do not have anti-derivatives that are expressible in terms of elementary functions. > This means we cannot find probabilities of a Normally distributed random variable by hand. > >So, what is the solution? >Use computers or tables of values. ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ] .right-column[ &nbsp; &nbsp; The use of tables for evaluating normal probabilities depends on the following relationship. If `\(X \sim \text{Normal}(\mu, \sigma^2)\)`, `$$\begin{align} P[a \le X \le b] &= \int\limits_a^b\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x - \mu)^2/{2\sigma^2}}dx \\ \\ &= \int\limits_{(a- \mu)/\sigma}^{(b-\mu)/\sigma}\frac{1}{\sqrt{2\pi}} e^{-z^2/2}dz \\ \\ &= P\left[\frac{a - \mu}{\sigma} \le Z \le \frac{b - \mu}{\sigma}\right] \end{align}$$` where `\(Z \sim \text{Normal}(0, 1)\)`. ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ ###Standard Normal Distribution The parameters are important in determining the probability, but because the pdf of a normal random variable is difficult to work with we often use the distribution with `\(\mu = 0\)` and `\(\sigma^2 = 1\)` as a reference point. >**Definition: Standard Normal Distribution** </br> >The standard normal distribution is a normal distribution with `\(\mu=0\)` and `\(\sigma^2=1\)`. It has pdf >$$ >\begin{align} >f(z) &= \dfrac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} z^2} \\\\ > &= \dfrac{1}{\sqrt{2 \pi}} \exp\left(-\frac{1}{2} z^2\right) \\\\ >\end{align} >$$` We say that a random variable is a "standard normal random variable" if it follows a standard normal distribution or that `\(Z \sim N(0, 1)\)`. ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ ### Standard Normal Distribution (cont) It's worth pointing out the reason why the standard normal distribution is important. There is no "closed form" for the cdf of a normal distribution. In other words, since we can't finish this step: $$ F(x) = \int_{-\infty}^{x} \dfrac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2 \sigma^2} (t - \mu)^2} dt = ??? $$ we have to estimate the value each time. However, we have already done this for _standard_ normal random variables already in **Table B.3** So if `\(Z \sim N(0, 1)\)` then `\(P(Z \le 1.5) = F(1.5) = 0.9332\)`. The good news is that we can connect any normal probabilities to the values we have for the standard normal probabilities. ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ ### Standard Normal Distribution (cont) These facts drive the connection between different normal random variables: >**Key Facts: Converting Normal Distributions**</br> > If `\(X \sim N(\mu, \sigma^2)\)` and `\(Z = \dfrac{X - \mu}{\sigma}\)` then `\(Z \sim N(0, 1)\)` > &nbsp;</br> > &nbsp;</br> > If `\(Z \sim N(0, 1)\)` and `\(X = \sigma Z + \mu\)` then `\(X \sim N(\mu, \sigma^2)\)` We use this connection as a way to avoid working with the normal pdf directly. ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Standard Normal ] .right-column[ ### Standard Normal Distribution (cont) A rule of thumb in dealing with questions about finding probabilities of Normally distributed probabilities of `\(N(\mu, \sigma^2)\)`: >(1) Translate that question to standard Normal distribution. i.e. `\(Z\sim N(0,1)\)` > >(2) Look it up in a table ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ ###CDF of Standard Normal Distribution The standard Normal distribution $ Z\sim N(0,1)$ plays an important rule in finding probabilities associated with a Normal random variable. The **CDF** of a standard Normal distribution is $$ \Phi(z) = F(z) = \int\limits_{-\infty}^z\frac{1}{\sqrt{2\pi}}e^{-t^2}dt = P(Z \leq z) . $$ Therefore, we can find probabilities for all normal distributions by tabulating probabilities for only the standard normal distribution. We will use a table of the **standard normal cumulative probability function**. ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ ### Standard Normal Distribution (cont) **Example: Normal to Standard Normal** If `\(X \sim N(3, 4)\)` then: $$ `\begin{align} P(X \le 6) &= P\left(\frac{X - 3}{2} \le \frac{6 - 3}{2} \right)\\\\ &= P(Z \le 1.5) \\\\ &= 0.9332 \end{align}` $$ where the valeu 0.9332 if found from **Table B.3** ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ ### Standard Normal Distribution (cont) **Example**: Standard normal probabilities `\(P[Z < 1.76]\)` &nbsp; &nbsp; `\(P[.57 < Z < 1.32]\)` ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ <center> <img src="normal_table1.jpg" alt="normal_table1.jpg" width="600" height= "550"/> </center> ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ <center> <img src="normal_table2.jpg" alt="normal_table2.jpg" width="600" height= "550"/> </center> ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ ### Some useful tips about standard Normal distribution >By symmetry of the standard Normal distribution around zero > `\(P(Z\ge a)= P(Z\le -a)\)` &nbsp; &nbsp; &nbsp; >We can also do it reverse, find `\(z\)` such that `\(P(-z \le Z \le z)= 0.95\)` > > `\(P(Z \geq \#)= 0.025\)` ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ **Example**: Baby food J. Fisher, in his article Computer Assisted Net Weight Control (**Quality Progress**, June 1983), discusses the filling of food containers with strained plums and tapioca by weight. The mean of the values portrayed is about `\(137.2\)`g, the standard deviation is about `\(1.6\)`g, and data look bell-shaped. Let $$ W = \text{the next fill weight.} $$ Let `\(W\sim N(137.2, 1.6^2)\)`. Find the probability that the next jar contains less food by mass than it's supposed to (declared weight = `\(135.05\)`g). ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ ### More example Using the standard normal table, calculate the following: `\(P(X > 7), X \sim \text{Normal}(6, 9)\)` &nbsp; &nbsp; &nbsp; `\(P(|X - 1| > 0.5), X \sim \text{Normal}(2, 4)\)` ] --- layout:false .left-column[ ## Background ## Terms and Use ## Common Dists ### Uniform ### Exponential ### Normal ### Std. Normal ] .right-column[ ### More example Find `\(c\)` such that `$$P(|X - 2| > c) = 0.01$$` where `\(X \sim \text{Normal}(2, 4)\)` ]