# A Concise Introduction to Statistical Inference

This module introduces concepts of statistical inference from both frequentist and Bayesian perspectives. Lesson 4 takes the frequentist view, demonstrating maximum likelihood estimation and confidence intervals for binomial data. Lesson 5 introduces the fundamentals of Bayesian inference.

This framework is extended with the continuous version of Bayes theorem to estimate continuous model parameters, and calculate posterior probabilities and credible intervals. Enroll for Free. From the lesson. Lesson 4.

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Introduction to R Plotting the likelihood in R Panik See All. Linear Programming and Resource Allocation Modeling. Stochastic Differential Equations. Introduction to Quantitative Methods in Business. Growth Curve Modeling. Advanced Statistics from an Elementary Point of View. The exponential distribution is used to model the lifetimes of electronic components and the waiting times between rare events. Gamma Distribution. The Beta Distribution.

## A Concise Introduction to Statistical Inference

We write f as fX,Y when we want to be more explicit. Let X, Y be uniform on the unit square. Random Variables 2. Here is an example which I borrowed from DeGroot and Schervish The trick here is to be careful about the range of integration. We pick one variable, x say, and let it range over its values.

It may help if you look at Figure 2.

### Course Prequisites

You can see this by drawing a diagram. The density is positive over this region. Suppose that fX,Y is given in the table that follows. The marginal distribution for X corresponds to the row totals and the marginal distribution for Y corresponds to the columns totals. Fortunately, we have the following result which we state for continuous random variables though it is true for discrete random variables too.

Let X and Y have joint pdf fX,Y. Let X and Y have a joint uniform distribution on the unit square. This can sometimes be useful as in example 2. Random Variables of Y? Consider the density in Example 2. We call X a random vector. Let f x1 ,. We say that X1 ,. If X1 ,. If F has density f we also write X1 ,. We also call X1 ,. Much of statistical theory and practice begins with iid observations and we shall study this case in detail when we discuss statistics.

The multivariate version of a Binomial is called a Multinomial. The marginal distribution of Xj is Binomial n,pj. Multivariate Normal. Random Variables where Z1 ,.

How do we compute the pdf and cdf of Y? In the discrete case, the answer is easy. The set Ay for example 2. Plot the cdf F. Let X and Y be independent and suppose that each has a Uniform 0, 1 distribution. Find the density fZ z for Z. Let X have cdf F. Let X and Y be independent. Show that g X is independent of h Y where g and h are functions. Suppose we toss a coin once and let p be the probability of heads. Let X denote the number of heads and let Y denote the number of tails.

Let X and Y be the number of heads and tails. Show that X and Y are independent. Prove Theorem 2. Plot it. Draw a histogram of y and compare it to the pdf you found in part a. Find the cdf and pdf of R. A universal random number generator. Let X have a continuous, strictly increasing cdf F. Find the density of Y.

## All of statistics: A Concise Course in Statistical Inference by Larry Wasserman - Ai Kenya

This is called the probability integral transform. Solve the following using the Normal table and using a computer package.

Prove formula 2. Let X1 ,. Find the pdf of Y. The expectation is a one-number summary of the distribution. Flip a fair coin two times. 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. How do we compute E Y?

But there is an easier way. Here is 3.

A In other words, probability is a special case of expectation. Let X, Y have a jointly uniform distribution on the unit square. What is the mean of X? Make sure you see why this makes intuitive sense. Expectation 3.

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If X and Y are random variables, then the covariance and correlation between X and Y measure how strong the linear relationship is between X and Y. The calculations for some of the others are in the exercises. Here is a subtle point. This is a very confusing point so let us look at an example. Consider example 3. How can we compute E Y? An easier way is to do this in two steps. Draw a county at random from the United States. Then draw n people at random from the county. Let X be the number of those people who have a certain disease.

- A Concise Introduction To Statistical Inference.
- Introduction to Statistical Inference.
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If Q denotes the proportion of people in that county with the disease, then Q is also a random variable since it varies from county to county. Suppose that the random variable Q has a Uniform 0,1 distribution. Let us compute the variance of X. This 3.