Math 52 — Notes on Probability(These are sketchy notes to accompany our discussion in lecture and section, and thetext pages 361-2. Please let us know of any typos.)1. Random Variables and PDFs in R1.We say a function p(x) is a PDF (probability density/distribution function) if p(x) ≥ 0for all x, andRRp(x) dx = 1.Consider a random occurrence with an outcome we call X, some real number. We say“X is a random variable with distribution p(x)” if for every a, b the probability that X liesbetween a and b isProb(a ≤ X ≤ b) =Zbap(x) dx.The interval [a, b] ⊂ R is also called an event. Notice that the only events we measure aresets (usually ranges) of values of X, not individual values of X.[Think of the x-axis as the set of possible values of the random variable X, and thegraph of p(x) as the continuous analogue of a histogram. This falls in line with p(x) havingan interpretation as a probability density, as the density principle would then dictate thatthe probability that X falls in a small range of width ∆x near X = x0is approximatelyp(x0)∆x; furthermore, probability is an aggregating quantity (like mass or area), so we mayadd together disjoint measurements and approximate the total probability via a Riemannsum. I n the limit as ∆x → 0, the Riemann sum approaches the above integral.]Expectation. The mean value or expected value of X, written E(X), is defined to bethe first moment of x with respect to p(x):E(X) =ZRxp(x) dx.(Note this corresponds to a weighted average of x, since E(X) =E(X)1=RRxp(x) dxRRp(x) dx.)In general, the expected value of some function f of X is the first moment of f(x), i.e.E(f (X)) =ZRf(x)p(x) dx.(For example, if X is the random variable giving the no ontime temperature, in degreesFahrenheit, on a January day in Palo Alto, then E(X) is the expected value of this tem-perature, and E(59(X − 32)) is the expected value of this temp erature measured in degreesCelsius.)Class e xerci se. Show that expectation value is linear, i.e. that for constants c and d,E(cf (X) + dg(X)) = cE(f(X)) + dE(g(X)).1Variance and Standard Deviati on. The variance of a random variable X is thesecond moment of X − E(X), i.e.Var(X) = E((X − E(X))2).The variance is also called the “second moment of X about the mean.” The standard devia-tion of X is the square root of Var(X).Class e xerci se. Show that Var(X) = E(X2) − E(X)2.Examples of PDFs1a. For constants σ > 0 and µ, letp(x) =1σ√2πe−(x−µ)2/2σ2,which is called the normal (Gaussian, bell-shaped) distribution with center µ and width σ.(You can use single-variable calculus to verify that the graph of p is symmetric about x = µ,has one local max at x = µ, and has two inflection points at x = µ ± σ.)By calculating the area under the curve using a trick, we proved in class last week thatin the case µ = 0, σ = 1, then p(x) is indeed a PDF. (See also Problem 33 on page 361.)Exercise: verify this for arbitrary values of µ, σ.Other facts: if X is a random variable with the above PDF, then E(X) = µ and Var(X) =σ2. Prove this! (Hint: to compute the integrals, first make the substitution t = (x − µ)/σ.)Furthermore, anyone who’s taken a stats course knows thatProb(µ − σ ≤ X ≤ µ + σ) ≈ 0.68, and Prob(µ − 2σ ≤ X ≤ µ + 2σ) ≈ 0.95.(You need a calculator to get these approximations, but how can you use the integrals totell that the quantities don’t depend on µ and σ?)1b. For a constant λ > 0, letp(x) =(1λe−x/λx ≥ 00 x < 0This is usually called the exponential-type distribution with width λ. It is often used tomodel the time spent waiting in a queue, or perhaps the lifespan of a light bulb.Exercises: show that p(x) is a PDF! What are the mean and the variance?1c. For any a, b, with a < b, the uniform distribution on [a, b] isp(x) =(1b−aa ≤ x ≤ b0 otherwiseYou should check that p(x) is a PDF, and compute the mean and variance. If X is a randomvariable with this distribution, what is the probability that X lies in [a, b]? outside thisinterval?22. Random Variables in R2and Joi nt PDFs.A two-variable function p(x, y) is a joint PDF (or 2-dim PDF) if p(x, y) ≥ 0 for all (x, y),andRRR2p(x, y) dA = 1.Consider a random occurrence with an outcome some ordered pair−→X ∈ R2. We say−→Xis a random variable with distribution p(x, y) if for every region D ⊂ R2,Prob(−→X ∈ D) =ZZDp(x, y) dA.Analogously to the R1case, the region D ⊂ R2is also called an event.For a random variable−→X = (X, Y ) with joint distribution p(x, y), the expectation (mean)value of some function f of X and Y is computed analogously to the R1case, as the firstmoment of f:E(f (X, Y )) =ZZR2f(x, y)p (x, y) dA.In particular, one could ask for the expected value of X, or of Y , alone, etc.Examples (via Text Problems)2a. Throw a dart at a dartboard; what are the (x, y) coordinates of the spot wherethe dart lands? (See also Problem 41: given a joint distribution on (x, y), compute theprobability that the dart lands inside a given region in R2, etc.)2b. Two lightbulbs, A and B; say the lifespan of bulb A is X hours and the lifespan ofbulb B is Y hours, each of which are random variables. We could package this informationtogether into a single 2-dimensional random variable−→X = (X, Y ). See also Problem 42: ifa single bulb has PDF p(x), which is an exponential-type distribution with λ = 2000, undercertain circumstances (independence; see below) it makes sense to say that the joint PDFof two bulbs is p(x, y) = p(x)p(y). What is the probability that both bulbs fail within 2000hours? What is the probability that bulb A fails before bulb B, but both fail within 1000hours?Marginal Probabilities and IndependenceGiven a random variable−→X = (X, Y ) in R2, it makes sense that we might want to focusour attention on X alone, or on Y alone, as examples of random variables in R1. The basicquestion is, given the joint PDF p(x, y) for−→X , what would be the PDF for either X aloneor Y alone? These are the marginal distributions: letpx(x) =ZRp(x, y) dy, and py(y) =ZRp(x, y) dx.Then px(sometimes written p1) is the single-variable PDF for X alone, and py(sometimeswritten p2) is the single-variable PDF for Y alone. (Why are pxand pynecessarily PDFs?)We say the two random variables X and Y in R1are independent if the joint PDF p(x, y)for the random variable−→X = (X, Y ) satisfies the property thatp(x, y) = px(x)py(y),3where pxand pyare the marginal distributions. (See also Problem