session 5
Post on 12-Nov-2014
161 Views
Preview:
DESCRIPTION
TRANSCRIPT
7/4/2012
1
Characteristics of discrete probability distribution
Expected value, standard deviation of a random variable
Binomial Probability Distributions
Clear Tone Radios
Probability
Distribution
Discrete Continuous
Distribution of Number of separate enquiries leading to
new business proposals on any working day
x P[X=x]
0 0.20
1 0.35
2 0.25
3 0.10
4 0.05
5 0.05
total 1.00
Prob. dist. of no. of qualifying
enquiries in a day
0.00
0.10
0.20
0.30
0.40
0 1 2 3 4 5
no. of queries
probability
The probability distribution should depict/provide all possible
Information regarding the random variable in question.
32
Properties
• f(x) = P[X=x]; 0 ≤ f(x) ≤ 1; Σ f(x) =1
• CDF F(x) = P[X ≤x] = Σy ≤ x f(y)
• F is a step function with jumps only at the possible values
• Density to distribution function and vice versa
x F(x)
0 0.20
1 0.55
2 0.80
3 0.90
4 0.95
5 1.00
Expectation and Variance
x
valuesf
prob.
x f x2f (x-mean)2f
0 0.20 0 0 0.512
1 0.35 0.35 0.35 0.126
2 0.25 0.5 1 0.04
3 0.10 0.3 0.9 0.196
4 0.05 0.2 0.8 0.288
5 0.05 0.25 1.25 0.578
Total 1.00 mean= 1.60 4.30 σ2=1.74
Check: 4.30 - 1.602= 1.74
Interpretation of
EXPECTED VALUE
• “mean” in the long run
• not necessarily the most likely value
7/4/2012
2
Probability distribution of # of successful days in a working
week (consisting of six days)
A day is deemed to be successful if at least 1 qualifying enquires
are made on that day.
P(a day is successful) = 0.8
Binomial Probabilities
Possible number of successful days : 0,1,2,3,4,5,6
Probability[0 successful day] = P[FFFFFF]=0.26=0.000064
Probability[1 successful day] = P[SFFFFF]+ P[FSFFFF]+
P[FFSFFF]+ P[FFFSFF]+ P[FFFFSF]+ P[FFFFFS]
= 6 ×0.25 ×0.8 =0.001536
Probability distribution of # of successful days in a
working week (consisting of six days):
Binomial distribution
6 4 2
4 0.8 0.2C × ×
21
562
6
4
6
×
×== CC
No of
successful days Probability
0 0.000
1 0.002
2 0.015
3 0.082
4 0.246
5 0.393
6 0.262
Binomial Distribution
When is it applicable?
• Binomial expt. is one where n independent
and identical trials are repeated; each trial
may result in two possible outcomes(call them
‘success’(S) and ‘failure’(F); p=P(S).
• In the above context, a random variable X,
denoting the total no. of successes is said to
have a Binomial distribution with parameters
n and p. X→B(n,p)
Binomial Distribution(cont.)
• So, in general, for X→B(n,p)
– P[X=x]= nCx px (1-p)(n-x)
• Use Excel to calculate probabilities
• Mean or the ‘Expected value’ = np
• Variance = np(1-p)
• standard deviation = √{np(1-p)}
40
Exercise
Work out the market research problem for the
new product design given in the last week.
Need to feed in P [survey result| Market will do
well] & P [survey result| Market will NOT do
well]
7/4/2012
3
43
Redo Problem (using tables)
•Let X denote the no. of transmissions with flaws(F).
•X→B(10,.02)
(a) want P[X>2] = 0.0008
(from AS: Appendix C)
(a) P[X= 0] = 0.8171
top related