The Normal Probability Curve

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{6}A_{64} Base Sequence = 12735 String Sequence = 12735 - 6 - 64 **

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The Normal Probability Curve

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(a) Suppose that the frequencies of some data is normally distributed and figure 118.5 represents the probability curve. What is the probability of a value occurring between *a* and *b*?

(b) The weight of a large number of grapefruits were found to be normally distributed with a mean of 1 lb and a standard deviation of 3 oz. What is the probability that any one grapefruit has a weight between 1 lb 3 oz and 1 lb 6 oz?

(c) The average number of persons joining a certain queue in one minute is 2. What is the probability that 5 persons will join the queue in one minute?

**The strings**:
S_{7}P_{6}A_{64} (Grouping - Multi-criteria).
**The math**:

Pj Problem of Interest is of type *grouping* (multi-criteria). *Grouping* is at the heart of statistics. The grouping may be permutational or combinational, single criterion or multi-criteria grouping.

(a) The probability that a value lies within a and b is given by the area under the curve between a and b

So, The probability that a value lies within a and b = ∫_{a} ^{b} f(x) dx.

Where f(x) is as indicated in figure 118.5

Mean = arithmetic average = [f_{1}x_{1} + f_{2}x_{2} ... f_{n}x_{n}]/n

Where f_{i} is the number of x_{i} in the data. i = 1, 2,...n.

Standard deviation σ = [Σ_{i} [f_{i}(x_{i} - mean)^{2}]/n]^{1/2}. i = 1, 2,...n.

(b) Normal frequency curves are varied. However, they are all characterized by their mean and standard deviation. Irrespective of the value of the mean and standard deviation, 68.2 % of the data lie within one standard deviation (σ) on either side of the mean; 95.4 % of the data lie within 2σ on either side of the mean and 99.8 % lie within 3σ on either side of the mean.

The interval of interest is between σ and 2σ to the right of the mean.

So, desired probability = 0.954/2 - 0.682/2 = 0.477 - 0.341 = 0.136.

(c) The *Poisson distribution* is the pertinent probability distribution of interest

The *Poisson distribution* says that if the mean number of events of a particular type in a fixed time interval is μ, then
the probability of n events p(n) occurring in one interval is given by:

p(n) = (μ^{n}e^{-μ})/ n! Where n! is n factorial = n(n-1)(n-2)...1

So, p(5) = (2^{-2})/(5x4x3x2x1) = [32/(2.718)^{2}]/120 = 0.036.

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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