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An excerpt from a university textbook on probability theory, specifically focusing on the binomial distribution. It covers the definition of a binomial experiment, the properties of such experiments, and the calculation of probabilities using the binomial distribution. The document also includes examples and exercises for practice. The binomial distribution is important in statistics as it models the number of successes in a fixed number of independent trials with a constant probability of success.
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STAT 3401, Intro. Prob. Theory 23/27 Jaimie Kwon
♦ Definition 3.6 A “binomial experiment” possesses the following properties: à A fixed number, n, of trials à Each trial results in one of two outcomes, Success and Failure (or S and F) [dichotomous or binary] à Each trial has the same success probability, p [identical] à The trials are independent [independence] ♦ Let Y=# of successes observed during n trials ♦ Example 3.5 and 3.6. Y1 = # of radar units (out of 4) that do not detect the plane Y2 = # of persons favoring a candidate in a random sample of n=10 voters Do they follow binomial distribution? ♦ Conditional/unconditional probabilities ; à The notion of hypergeometric distribution ♦ Definition 3.7 A random variable Y is said to have a “binomial distribution” based on n trials with success probability p if and only if p(y) = nCy p yq n-y^ , y=0,1,…,n and 0≤p≤ 1 à Histograms? (Use R) ♦ Example 3.7 5000 fuses contain 5% defectives; for a sample of 5 fuses, what’s P(Y≥1)=? How about P(Y≥4)=? ♦ Notation: Y~bin(n,p) ♦ Theorem 3.7 Let Y~bin(n,p). Then, μ=E(Y)=np and σ^2 =V(Y)=npq à Proof: definitions and ‘binomial’ trick ♦ In practical application, carefully define ‘successes! ♦ HW. Some of the exercises 3.25~ ♦ Keywords: binomial experiment; Y~bin(n,p)