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Geometric and Related Probability Distributions - STAT 3401, Assignments of Probability and Statistics

The geometric probability distribution, negative binomial distribution, and hypergeometric distribution. The geometric distribution deals with the number of trials until the first success, while the negative binomial distribution deals with the number of trials until the r-th success. The hypergeometric distribution is similar to the binomial distribution but considers sampling without replacement. Exercises and examples using r.

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Pre 2010

Uploaded on 08/19/2009

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STAT 3401, Intro. Prob. Theory 24/29 Jaimie Kwon
1/27/2005
3.5 The geometric probability distribution
Same setup as binomial experiments
Y=(# of trials on which the first success occurs)
Possible values of Y?
E1: S,
E2: FS,
E3: FFS,
Ek: FFF…FFFS (k F’s)
p(y)=P(Y=y)=P(E
y
)=P(FFF…FFFS)=qqq…qqqp = q
y-1
p
Definition 3.8 A random variable y is said to have a geometric probability distribution iff
p(y)=qk-1p, y=1,2,3,… 0p1.
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(Y2)
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)/p2
à Proof:
What’s E(Y) and var(Y) for the Y above?
HW. Some of the exercises 3.50~71
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
,...2,1,,
1
1
)( ++=
=rrryqp
r
y
yp ryr
Also,
µ=E(Y)=r/p and σ
2
=V(Y) = r(1-p)/p2
pf2

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Download Geometric and Related Probability Distributions - STAT 3401 and more Assignments Probability and Statistics in PDF only on Docsity!

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)