Transcript Chapter 8

Chapter 8
Binomial and Geometric Distributions
8.1 The Binomial Distributions
• Binomial Setting
– Each observation falls into one of two categories,
success or failure
– There is a fixed number n of observations
– The n observations are all independent
– The probability of success p is the same for all
observations.
• ALL FOUR MUST BE MET to be BINOMIAL
Binomial Distribution
• The distribution of the count X of successes in
the binomial setting is the binomial
distribution with parameters n and p.
– The parameter n is the number of observations,
– and p is the probability of a success on any one
observation.
– The possible values of X are the whole numbers
from 0 to n.
– B(n, p)
PDF
• Given a discrete random variable X, the PDF –
probability distribution function – assigns a
probability to each value of X
– Must satisfy all probability rules
• Binompdf (n, p, x) command is found under
2nd DISTR/ 0: binompdf on TI 83
• EX 8.5
CDF
• Given a random variable X, the CDF –
cumulative distribution function – of X
calculates the sum of the probabilities for
0,1,2,…, up to the value X. It calculates the
probability of obtaining at most X successes in
n trials.
Binomial Coefficient
• The number of ways of arranging k successes
among n observations is given by the binomial
coefficient
For k = 0, 1, 2, …., n
8.3: X=type O blood, n=5, p=0.25
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B(5, 0.25)
BPDF(5, .25, 2) = 0.2637
BPDF(n, p, # of succ)
X
0
1
2
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Bpdf
Bcdf
C. Verified in CDF
4
5 L1
L2=binpdf(5,.25)
L3=bincdf(5,.25)
Histogram of PDF
Histogram of CDF
Binomial Probability
• If X has a binomial distribution with n
observations and probabilities p of success on
each observation, the possible values of X are
0,1,2,….,n. If k is any one of the values
• #2, 5, 7, 8, 12, 13
• In class 3, 4, 6, 9, 10, 11
8.1 part 2
• Mean and Standard deviation of a binomial
random variable
– If a count X has a binomial distribution with
number of observations n and probability of
success p, the mean and standard deviation of X
are
– μ=np
σ=√np(1-p)
Normal approximation for binomial
distributions
• Suppose that a count X has a binomial dist.
With n trials and success probability p.
• When n is large, the distribution of X is
approximately normal, N(np, √np(1-p))
• As a rule of thumb we will you the normal
approximation when n and p satisfy
– np≥10 and
– n(1-p)≥10
• Accuracy of N(np, √np(1-p)) improves as n gets
larger
• Most accurate for any fixed n when p is close
to ½
• Least accurate for any fixed n when p is close
to 0 or 1
• HW#17, 19b-d, 20, 26
• In Class#15, 16
8.2:Geometric Distributions
• Geometric Setting
– Each observation falls into one of two categories,
success or failure
– The probability of success p is the same for each
observation
– Observations are independent
– The variable of interest is the number of trials
required to obtain first success
Rule for calculating geometric
probabilities
• If X has a geometric distribution with prob p of
success and (1-p) of failure on each
observation, the possible values of X are 1, 2,
3, … If n is any one of these values, the prob.
that the first success occurs on the nth trail is
– P(X=n) = (1-p)n-1p (Formula)
– GPDF(p, n) GCDF(p, n)
n=1st success
Mean and Standard Deviation of
Geometric Random Variable
• μ = 1/p
• (1-p)/p2
• √(1-p)/p
mean
variance of X
standard deviation of X