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ST 380 Probability and for the Physical Sciences Statistics and Their Distributions

Suppose that several random variables X1, X2,..., Xn are associated with the outcome of some .

Then any function of them, such as their X + X + ··· + X X¯ = 1 2 n n is also a : that is, its value is unknown before the experiment is carried out, but is determined by the outcome of the experiment.

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Statistic Any quantity that can be calculated from the observed values of X1, X2,..., Xn without knowing their distributions is called a statistic.

Their mean X¯ is a statistic. If all the X s have µ, that is

E(Xi ) = µ, i = 1, 2,..., n, ¯ then (X − µ) is a function of X1, X2,..., Xn, but it is not a statistic, because we need to know µ, a of their distributions, to calculate it.

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Sampling Distribution of a Statistic A statistic is a random variable, and therefore has a .

The probability distribution of a statistic is called its distribution. To some extent, this is just terminology: a statistic is a random variable, and consequently has a probability distribution. We use the term to emphasize that statistics vary from to sample.

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Example 5.19

X1, X2,..., X6 are measured strengths of 6 randomly chosen items. Each has the Weibull distribution with α = 2, β = 5.

The pdf: curve(dweibull(x, 2, 5), to = 15)

¯ Simulate X1, X2,..., X6 and calculate X and S, the : x = rweibull(6, 2, 5); c(x, mean = mean(x), sd = sd(x))

Repeat: every sample is different, and so are the mean and sd.

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Simple Random Sample

Often, as in Example 5.19, the random variables X1, X2,..., Xn satisfy: They are independent; They all have the same distribution. In this case, we say that they are a (simple) random sample from that distribution.

Many statistical problems can be solved by finding the sampling distribution of some statistic in the context of a simple random sample.

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Deriving a Sampling Distribution The simplest case is the binomial experiment, with ( 1 if i th trial is a success Xi = . 0 otherwise .

P ¯ Then X = Xi ∼ Bin(n, p), and X = X /n, so

n P(X¯ = x/n) = P(X = x) = px (1 − p)n−x , x = 0, 1,..., n. x

In some similar special cases, we can find the distribution of a statistic of interest exactly.

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Simulation In many cases, such as the Weibull samples of Example 5.19, the sampling distributions of the mean and the standard deviation cannot be written in any simple form.

We can always approximate these distributions by simulation (a simple form of the ): nsim <- 1000 n <- 6 x <- matrix(rweibull(nsim * n, 2, 5), nsim, n) hist(apply(x, 1, mean)) hist(apply(x, 1, sd))

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We created nsim = 1000 samples, each of size n = 6, and calculated the mean and sd of each.

For any x > 0, the fraction of sample that are ≤ x is close to ¯ P(X ≤ x) = FX¯ (x), and gets closer as nsim is made larger.

More generally, we can get a good approximation to any probability involving the sample mean, the sample sd, or any other statistic, by making nsim large enough.

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