Binomial Distribution

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Binomial Distribution (Page 145 of Text)
Binomial distribution applies to cases where
there are only two possible outcomes head or
tail, success or failure and defective item or
good item.
Requirements for using Binomial Distribution
Outcome is determined completely by chance.
There are only two possible outcomes.
All trials have the same probability for a
particular outcome.
The number of trials must be fixed.
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Binomial Distribution
Let p probability that an item is defective q
probability that an item is good The item can
either be good or defective. Therefore, q 1-p
If n items are produced, find the probability of
exactly r defective items.
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Binomial Distribution
The number of trials, n, is fixed in this case.
We have to find the probability that r items are
defective and (n r) items are good.
But r defective items and (n r) good items can
be arranged in various ways.
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Binomial Distribution
r defective items and (n r) good items can be
arranged in nCr ways.
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Binomial Distribution
A sample of eleven electric bulbs is drawn every
day from those manufactured at a plant.
Probabilities of defective bulbs are random and
independent of previous results. The probability
that a bulb is defective is 0.04.
What is the probability of finding exactly three
defective bulbs in a sample?
What is the probability of finding three or more
defective bulbs in a sample?
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Binomial Distribution
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Binomial Distribution
Pr3 defective bulbs 0.0076
Pr3 or more def. 1- Pr0 def. Pr1 def.
Pr2 def.
1 0.6382 - 0.2925 0.0609 0.0084
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Binomial Distribution
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Binomial Distribution
Expectation and Standard Deviation
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Binomial Distribution
Variance
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Binomial Distribution
Standard Deviation
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Binomial Distribution
For p 0.35 and n 10
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Probability Distributions of Continuous
Variables (Page 184 of Text)
It is not possible to count the number of
possible values of a continuous variable.
The probability that the value of a continuous
random variable will be between limits a and b is
given by an integral.
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The function f(x) is called a probability density
function. The probability that the continuous
random variable, X, is between a and b
corresponds to the area under the curve
representing f(x).
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Probability for a Continuous Random Variable
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The Uniform Distribution
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The Exponential Distribution
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Expected Value
For a continuous random variable the expected
value is
Note that
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Variance
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A probability density function is given by
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A Probability Density Function
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• Find the probability that X is between 1 and 2.
• Find the cumulative distribution function of X.
• Find the expected value of X.
• d) Find the variance and standard deviation of
  X.

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• Find the probability that X is between 1 and 2.

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(b) Find the cumulative distribution function of
X.
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(b) Find the cumulative distribution function of
X.
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The cumulative distribution function is
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• Find the expected value of X.

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d) Find the variance and standard deviation of
X.
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d) Find the variance and standard deviation of
X.
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