By Anderson, Stewart
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Extra resources for Biostatistics
3% of the time. , X∼ ˙ Pois(λ = np). Note also that if q ≈ 1, then npq ≈ np yielding a distribution whose mean and variance are approximately equal such as in the Poisson distribution. 12. 01), respectively. The first eight values (j = 0, 1, . . 2. 002 in the second approximation. 01 and n ≥ 100. 2. 8 The Chi-Squared Distribution If Z ∼ N (0, 1), then Z 2 ∼ χ21 . The sum of the squares of independent standard normal random variables is a chi-squared distribution, that is, if Z1 , Z2 , . . , Zr are independently and identically distributed as N (0, 1) then S 2 = Z12 + Z22 + .
The degrees of freedom can be thought of as pieces of information. The estimate of the sample standard deviation is given by n 2 s = j=1 (xi − x) /(n − 1) so that once the mean, x, is calculated then only n − 1 pieces of information are needed to calculate s. Each of the members of the family of central t-distributions look very similar to the standard normal distribution except for the fact that the t-distributions have wider tails than does the standard normal distribution. It can be shown that as n → ∞, tn−1 → N (0, 1).
A 2 × 2 table summarizing this scenario is shown below. Group 1 Group 2 Fail a b a+b Fail c d c+d a+c b+d N Assuming that we know that a + b subjects fail out of N = a + b + c + d patients, we can calculate the probability of a out of a + c patients failing in population #1 and b out of b + d subjects failing in population #2 as a+c a b+d b N a+b = (a + c)! (b + d)! (a + b)! (c + d)! N ! a! b! c! d! This distribution has expected value (a+b)(a+c) N and variance (a+b)(a+c)(c+d)(b+d) . N 2 (N −1) THE HYPERGEOMETRIC DISTRIBUTION 37 The hypergeometric distribution is used in statistical applications to calculate what is known as Fisher’s exact test.
Biostatistics by Anderson, Stewart