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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.

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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

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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)}

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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]

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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