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All of Statistics: A Concise Course in Statistical Inference by Larry Wasserman

By Larry Wasserman


This booklet is for those that are looking to research likelihood and records fast. It brings jointly the various major rules in sleek data in a single position. The e-book is acceptable for college kids and researchers in facts, desktop technological know-how, facts mining and laptop learning.

This booklet covers a wider variety of issues than a standard introductory textual content on mathematical records. It comprises smooth subject matters like nonparametric curve estimation, bootstrapping and class, issues which are often relegated to follow-up classes. The reader is thought to grasp calculus and a bit linear algebra. No prior wisdom of chance and records is needed. The textual content can be utilized on the complex undergraduate and graduate point.

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37 Example. Let X and Y have a joint uniform distribution on the unit square. Thus, fXly(xly) = 1 for 0 <::: x <::: 1 and 0 otherwise. Given Y = y, X is Uniform(O, 1). • From the definition of the conditional density, we see that fx,Y(x, y) fXIY(xly)jy(y) = jylx(Ylx)fx(x). 39. 38 Example. Let f(x,y)= Let us find lP'(X y + (1/2). Hence, { < 1/41Y f X 0 +y if 0 <::: x <::: 1, 0 <::: y <::: 1 ' oth erWlse. 1/3). 27 we saw that jy(y) ( I ) - fx,y(x,y) XIY x y jy(y) So, 11/4 fxlY (x I 1 1/4 o X + 1.

THE BINOMIAL DISTRIBUTION. Suppose we have a coin which falls heads up with probability p for some 0 s: p s: 1. Flip the coin n times and let X be the number of heads. Assume that the tosses are independent. Let f(x) = J1D(X = x) be the mass function. It can be shown that f(x)= { (n) PX(l ox -p )n-x for x = 0, ... ,n otherwise. A random variable with this mass function is called a Binomial random variable and we write X rv Binomial(n,p). If Xl rv Binomial(nl,p) and X 2 rv Binomial(n2,p) then Xl +X2 rv Binomial(nl + n2,p).

If you simulate a Cauchy distribution many times and take the average, you will see that the average never settles down. This is because the Cauchy has thick tails and hence extreme observations are common. _ From now on, whenever we discuss expectations, we implicitly assume that they exist. Let Y = r(X). How do we compute lE(Y)? One way is to find Jy(y) and then compute lE(Y) = I yfy(y)dy. But there is an easier way. 6 Theorem (The Rule of the Lazy Statistician). Let Y = r(X). Then lE(Y) = lE(r(X)) = J r(x)dFx(x).

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