STATISTICS - CH 6, LECTURE
6.1
Probability Distributions
Notion of a random variable: a quantity that can take on different values
Ex: x = sum of spots on roll of two dice
Ex: x = nr heads on 4 flips of a coin
probability is a function of x
Use of Table X (binomial coeffs)
Prob must be between 0 and 1
Sum of all values of a prob dist must be 1
6.2 Binomial Distribution
Typical situations
Probability of 5 heads on 8 flips of a coin
Probability of getting three 6's on 15 rolls of one die
Formula: P(x) = f(x) = 
For the distrib, x =
0, 1, 2, 3, ..., to n
Interpretation
Application: Calculate for the three 6's on 15 rolls of one die
When to use?
Fixed nr of trials
Each trial is counted as a "success" or a "failure"
Probability of success is the same on each trial -- so applies to selection with replacement
Trials are independent
Ex: Problem 6.7
a. Calculate for one value of x. With exam designed so that 80% of people pass,
if 14 take the
test, what is prob that 12 will pass, i.e. P(x=12)?
b. Find P(x³12) = P(x=12) + P(x=13) + P(x=14) by the formula
c. Tedious, so use table I, pg 484
Distribution: Calculate for all possible values of x -- or, given in table
When can you not use the table?
n³16
p not listed (table has increments of 10%, and also 5%, 95%)
See # 18 & especially19 on probabilities of committing errors of rejecting claims or accepting claims.
6.3
Hypergeometric Distribution
Ex: situation
Prob of getting exactly 3 red m&m's if 7 are selected from a jar containing 8 reds and 20 others
We've done before:
Use table to show how total set is partitioned, and how many are chosen from each part
Formula -- pg 211
Ex: Calculate for one value of
x:
P(x=3) given a=8, b=20-8,
n=7 [same as example above]
Distribution: Calculate for all possible values of x, x=0, 1, 2, ..., 7
We can use binomial dist to approximate the hypergeometric,
if sample is no more than 5% of population (n £ .05 (a+b))
Example
6.4 Poisson Distribution (and Poisson Approximation to Binomial)
Formula to use Poisson Distrib as approximation to binomial dist:
Applicable when n ³ 10 and np < 10.
e is approximately 2.71828
On many calculators, enter 1 INV LN to get e.
Or to get ex for any exponent x, enter x INV LN
Return to: Merced College; Don Power Updated 03/05/04 by Don Power