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Binomial Probability Theory: Understanding the Binomial Distribution and Experiments, Assignments of Probability and Statistics

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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Pre 2010

Uploaded on 08/19/2009

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STAT 3401, Intro. Prob. Theory 23/27 Jaimie Kwon
1/26/2005
3.4 The binomial probability distribution
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 pyqn-y, y=0,1,…,n and 0p1
à Histograms? (Use R)
Example 3.7 5000 fuses contain 5% defectives; for a sample of 5 fuses, what’s P(Y1)=?
How about P(Y4)=?
Notation: Y~bin(n,p)
Theorem 3.7 Let Y~bin(n,p). Then,
µ=E(Y)=np and σ2=V(Y)=npq
à Proof: definitions and ‘binomial’ trick
In practical application, carefully define ‘successes!
HW. Some of the exercises 3.25~49
Keywords: binomial experiment; Y~bin(n,p)

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STAT 3401, Intro. Prob. Theory 23/27 Jaimie Kwon

3.4 The binomial probability distribution

♦ 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)