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STAT 3401, Intro. Prob. Theory 24/29 Jaimie Kwon
1/27/
3.5 The geometric probability distribution
♦ Same setup as binomial experiments
♦ Y=(# of trials on which the first success occurs)
♦ Possible values of Y?
♦ E 1 : S,
E 2 : FS,
E 3 : FFS,
…
Ek : FFF…FFFS (k F’s)
…
♦ p(y)=P(Y=y)=P(E y)=P(FFF…FFFS)=qqq…qqqp = q y-1p
♦ Definition 3.8 A random variable y is said to have a geometric probability distribution iff
p(y)=q k-1p, y=1,2,3,… 0≤p≤1.
♦ Do they add up to 1? (Exercise 2.50)
♦ Probability histogram (use R; barplot(dgeom(1:7,p=.5))
♦ Often used to model distributions of lengths of waiting times.
♦ Example 3.11 The probability of engine malfunction during any 1-hour period is p=.02. What’s
P(a given engine will survive 2 hours)=?
à Let Y=(# of 1-hour intervals until the first malfunction)
à P(survive 2 hours) = P(Y≥2)
♦ Theorem 3.8. If Y is a random variable with a geometric distribution,
μ=E(Y)=1/p and σ^2 =V(Y) = (1-p)/p
à Proof:
♦ What’s E(Y) and var(Y) for the Y above?
♦ HW. Some of the exercises 3.50~
♦ Keywords: geometric distribution; Y~geom(p)
3.6 The negative binomial probability distribution
♦ Y=the number of the trial on which the r’th success occurs
♦ Y~NB(r,p) has a probability function
− pq y rr r r
y p y
r yr
Also,
μ=E(Y)=r/p and σ^2 =V(Y) = r(1-p)/p
STAT 3401, Intro. Prob. Theory 25/29 Jaimie Kwon
1/27/
♦ In R, run barplot(dnbinom(1:10, 5, .5))
♦ Skip in the current class but it’s a distribution useful in many applications
♦ HW. Some of the exercises 3.72~
♦ Keywords: negative binomial distribution; Y~nbinom(r,p)
3.7 The hypergeometric probability distribution
♦ Similar to binomial experiment but samplingwithout replacement. (cf. voter poll example above)
Need to consider when n is large relative to N.
♦ Definition 3.10 A random variable Y is said to have a hypergeometric probability distribution if
and only if dist
n
N
n y
N r
y
r
p ( y ) where y = 0 , 1 ,..., n subject to y≤r and n-y ≤ N-r.
♦ Derivation?
♦ Example 3.16 Select 10 engineers randomly from a group of 20. What’s P(the 10 selected
include all the 5 best engineers)
♦ Theorem 3.10 For Y~hypergeometric, N
nr
μ= E ( Y )= and
2
N
N n
N
N r
N
r
σ V Y n.
♦ Resemblance to binomial distribution can be seen by letting p=r/N and letting N->∞.
♦ Example 3.17 Lots of 20. Reject a lot if more than 1 defective is observed out of 5 samples from
a lot. If a lot contains 4 defectives, what’s the probability that it will be rejected? What are the E
and V of the number of defectives in the sample of 5?
♦ HW. Some of the exercises 3.84~
♦ Keywords: hypergeometric distribution; Y~hyper(# of ‘success’ balls, # of ‘failure’ balls, sample
size)
