Welcome to MM207 Unit 4 Seminar Binomial and the Discrete Probability Function (w/ Excel)
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Transcript of Welcome to MM207 Unit 4 Seminar Binomial and the Discrete Probability Function (w/ Excel)
Definitions
• Statistical Experiment: Any process by which we obtain measurements or data.– In Unit 3 seminar we discussed dice. Rolling the dice is a
statistical experiment.• Random Variable: A random variable is the outcome of a
statistical experiment. We don’t know that this outcome will be before conducting the experiment– Discrete random variable: The possible values of the
experiment take on a countable number of results.• For the roll of a die there were 6 possible results.• Discrete variables have to be counted (like eggs)
– Continuous random variable: The possible values of the experiment are infinite.
• For example, measure the weight of 1 year old cows is continuous. The number of possibilities are uncountable. (500.12 pounds, 534.1534 pounds, etc…
• Continuous variables have to be measured (like weight or milk or gas)
Probability Distribution
• Probability distribution is the assignment of probabilities to specific values for a random variable or to a range of values for the random variable
• Plain English: Each random variable (outcome) from a random experiment has a particular probability of occurring.
• See page 196 Example 2
Score, x Probability, P(x)
1 0.16
2 0.22
3 0.28
4 0.20
5 0.14
Passive-Aggressive Traits
0
0.05
0.1
0.15
0.2
0.25
0.3
1 2 3 4 5
Score
Probability distribution properties
• Mean: This is the expected value of a probability distribution. This is the outcome about which the distribution is centered.
μ = ∑ x P(x)
• Standard deviation: This is the spread of the data around the expected value (mean)
σ = √ ∑ (x – μ)2 P(x)
• How do you use the equations? Let’s use EXCEL!!!!.
Excel Procedure for Mean & Standard Deviation
(Try It Yourself 5 & 6; pp. 198-199)
1 2 3 4 5 6
X P(X) X*P(X) X - MEAN (X-MEAN)^2 P(X)*(X-MEAN)^2
0 0.16 0 -2.6 6.76 1.0816
1 0.19 0.19 -1.6 2.56 0.4864
2 0.15 0.3 -0.6 0.36 0.054
3 0.21 0.63 0.4 0.16 0.0336
4 0.09 0.36 1.4 1.96 0.1764
5 0.1 0.5 2.4 5.76 0.576
6 0.08 0.48 3.4 11.56 0.9248
7 0.02 0.14 4.4 19.36 0.3872
1 2.6 3.72
MEAN VARIANCE
1.93
STANDARD DEVIATION
Features of the Binomial Experiment• Fixed number of trials denoted by n• n trials are independent and performed under
identical conditions• Each trial has only two outcomes: success
denoted by S and failure denoted by F• For each trial the probability of success is the
same and denoted by p. The probability of failure is denote by q and q = 1 - p)
• The central problem is to determine the probability of x successes out of n trials. P(x) = ?
Example
n = 10p = 0.4x = 6
Find P(x = 6)
Using the binomial table (Table 2 A8-A10)Using the binomial formulaUsing Excel
Using the Binomial Formula
P(x) = nCx px qn-x
x = number of successesn = number of trialsp = probability of one successq = probability of one failure (1 – p)
nCx is the binomial coefficient give by nCx = n! / [x! (n-x)!]
Remember 4! = 4*3*2*1 = 24 and is called factorial notation.
Using the Binomial Formulan = 10p = 0.4x = 6
Find P(x = 6)
P(x) = nCx px qn-x
nCx = 10C6 = 210
px = 0.46 = 0.004096qn-x = 0.610-6 = 0.64 =0.1296
210* 0.004096 * 0.1296 = 0.111476736 ≈ 0.111
Using Exceln = 10p = 0.4x = 6
Find P(x = 6)
Click on the cell where you want the answer.Under fx, find BINOMDIST
Number_s: Enter 6Trials: Enter 10Probability: Enter 0.4Cumulative: FalseP(x = 6) = 0.111476736 ≈ 0.111
Finding Cumulative Probabilitiesn = 10p = 0.4x ≤ 6
Find P(x ≤ 6)
Find each probability using the binomial table or the formula.
P(x ≤ 6) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6)
= 0.006 + 0.040 + 0.121 + 0.215 + 0.251 + 0.201 + 0.111 = 0.945
Use the complementP(x ≤ 6) = 1 – P(x > 6) = 1 – [P(x = 7) + P(x = 8) + P(x = 9) + P(x = 10)]= 1 – [0.042 + 0.011 + 0.002 + 0.000] = 1 – 0.055 = 0.945
Finding Cumulative Probabilities con’tn = 10p = 0.4x ≤ 6
Find P(x ≤ 6)
Use Excel• Number_s: Enter 6• Trials: Enter 10• Probability: Enter .4• Cumulative: TrueP(x ≤ 6) = 0.945238118 ≈ 0.945
Mean and Standard Deviation of the binomial probability distribution
• Mean or expected number of successμ = np
• Standard deviationσ = √ npq
• Where:n = number of trialsp = probability of successq = probability of failure (q = 1 – p)
Computing the Mean, Standard Deviation, and Variance for a Binomial Distributionn = 10p = 0.4
Meanμ = npμ = 10 * 0.4μ = 4
Standard deviation
σ = √ npq
σ = √ 10 * 0.4 * 0.6
σ = √ 2.4 ≈ 1.549
Varianceσ2 = npq = 2.4