Statistics 510: Notes 7 - Statistics Department



Statistics 510: Notes 14

Reading: Section 5.4

Note: Room and Time for Question and Answer Review Session for midterm. Monday, October, 16th, 6:30 pm, Huntsman Hall 265.

Midterm is in class on Wednesday.

I. Normal Random Variables

We say that X is a normal random variable or simply that X is normally distributed, with parameters [pic]and [pic]if the pdf of X is given by

[pic]

This density function is a bell-shaped curve that is symmetric about [pic].

[pic]

(a) smaller [pic]; (b) larger [pic].

The family of normal random variables plays a central role in probability and statistics. This distribution is also called the Gaussian distribution after Carl Friedrich Gauss, who proposed it as a model for measurement errors. The central limit theorem, which will be discussed in Chapter 8, justifies the use of the normal distribution in many applications. Roughly, the central limit theorem says that if a random variable is the sum of a large number of independent random variables, it is approximately normally distributed. The normal distribution has been used as a model for such diverse phenomenona as a person’s height, the distribution of IQ scores and the velocity of a gas molecule.

Linear transformations of normal random variables: An important fact about normal random variables is that if X is normally distributed with parameters [pic]and [pic], then [pic]is normally distributed with parameters [pic]and [pic]. To show this, suppose that [pic](the verification when [pic]is similar). Let [pic]denote the cdf of [pic]. Then

[pic]

where [pic]is the cdf of X. Differentiation yields that the pdf of [pic]is

[pic]

which shows that [pic]is normal with parameters [pic] and [pic].

An important implication of the preceding result is that if X is normally distributed with parameters [pic]and [pic], then [pic]is normally distributed with parameters 0 and 1. Such a random variable is said to be a standard normal random variable.

Mean and variance of normal random variables: We start by finding the mean and variance of the standard normal random variable [pic]. We have

[pic]

Thus,

[pic].

Integration by parts ([pic]) now gives

[pic]

Because [pic], the preceding yields the result

[pic]

It is traditional to denote the cdf of a standard normal random variable by [pic]. That is,

[pic]

The values of [pic]for nonnegative x are given in Table 5.1. For negative values of x, [pic] can be obtained from the equation

[pic],

which follows from the symmetry of the standard normal random variable.

Since [pic]is a standard normal random variable whenever X is normally distributed with parameters [pic] and [pic], it follows that the cdf of X can be expressed as

[pic].

Example 1: The following letter appeared in a well-known advice-to-the-lovelorn column:

Dear Abby: You wrote in your column that a woman is pregnant for 266 days. Who said so? I carried my baby for ten months and five days and there is no doubt about it because I know the exact date my baby was conceived. My husband is in the Navy and it couldn’t have possibly been conceived at any other time because I saw him only once for an hour, and I didn’t see him again until the day before the baby was born.

I don’t drink or run around, and there is no way this baby isn’t his, so please print a retraction about the 266-day carrying time because otherwise I am in a lot of trouble.

San Diego Reader

Let X denote the time of a pregnancy duration. According to well document norms, the mean and standard deviation for X are 266 days and 16 days, respectively. Assume the distribution of X is normal and calculate [pic].

Example 2: The army is developing a new missile and is concerned about its precision. By observing points of impact, launchers can adjust the missile’s initial trajectory, thereby controlling the mean of its impact distribution. If the standard deviation of the impact distribution is too large, though, the missile will be ineffective. Suppose the Pentagon requires that at least 95% of the missiles must fall within 1/8 mile of the target when the missiles are aimed properly. Assume the impact distribution is normal. What is the maximum allowable standard deviation for the impact distribution?

II. Normal Approximation to the Binomial Distribution

An important result in probability theory, known as the DeMoivre-Laplace limit theorem, states that when [pic]is large, a binomial random variable with parameters n and p will have approximately the same distribution as a normal random variable with the same mean and variance as the binomial.

DeMoivre-Laplace Limit Theorem:

If [pic]denotes the number of successes that occur when n independent trials, each resulting in a success with probability p, are performed, then for any [pic],

[pic]

as [pic].

Comments on normal approximation vs. Poisson approximation to binomial:

The normal distribution provides an approximation to the binomial distribution when [pic]is large [The normal approximation to the binomial will, in general, be quite good for values of [pic]satisfying [pic]].

The Poisson distribution provides an approximation to the binomial distribution when [pic]is large and [pic]is small so that [pic]is moderate.

Example 3: Airlines A and B offer identical service on two flights leaving at the same time (meaning that the probability of a passenger choosing either is ½). Suppose that both airlines are competing for the same pool of 400 potential passengers. Airline A sells tickets to everyone who requests one, and the capacity of its plane is 230. Approximate the probability that airline A overbooks.

Binomial approximation to hypergeometric (See Section 4.8.3). If n balls are randomly chosen without replacement from a set of N balls, of which the fraction [pic]is white, then the number of white balls selected is hypergeometric. When [pic]and N are both large in relation to n (say at least 50 times as large as n), it doesn’t make much difference whether the selection is being done with or without replacement. The number of white balls is approximately binomial with parameters [pic]and p. To verify this intuition, note that if X is hypergeometric, then for [pic],

[pic]

when [pic]and m and N are large in relation to n and i.

The fact that the binomial provides an approximation to the hypergeometric and the normal provides an approximation to the binomial can be combined to use the normal to provide an approximation to the hypergeometric when [pic]and m and N are large in relation to n and [pic].

Example 4: Suppose that sentiment for two political candidates for senator in Pennsylvania is split evenly. What is the probability that in a poll of 1000 randomly sampled voters, the proportion of voters preferring the Democratic candidate will be 0.55 or greater?

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