Applied Mathematics - Probability & Statistics

Description

Probability

• Random experiments −

  • Outcomes

  • Sample spaces (set representation)

• Events −

  • Occurrence of events, 'not', 'and' and 'or' events

  • Exhaustive events

  • Mutually exclusive events

  • Axiomatic (set theoretic) probability

  • Connections with the theories of earlier classes

• Probability of −

  • An event

  • probability of 'not', 'and' and 'or' events

Statistics

• Measures of dispersion −

  • Range

  • Mean deviation

  • Variance

  • Standard deviation of ungrouped/grouped data

• Analysis of frequency distributions with equal means but different variances.

SUMMARY

Probability

1. In this Chapter, we studied about the axiomatic approach of probability. The main features of this Chapter are as follows:

2. Sample space: The set of all possible outcomes

3. Sample points: Elements of sample space

4. Event: A subset of the sample space

5. Impossible event : The empty set

6. Sure event: The whole sample space

7. Complementary event or ‘not event’ : The set A′ or S – A

8. Event A or B: The set A ∪ B

9. Event A and B: The set A ∩ B

10. Event A and not B: The set A – B

11. Mutually exclusive event: A and B are mutually exclusive if A ∩ B = φ

12. Exhaustive and mutually exclusive events :-  Events E1 , E2 ,..., En are mutually exclusive and exhaustive if E1 ∪ E2 ∪ ...∪ En = S and Ei ∩ Ej = φ V i ≠ j

13. Probability : Number P (ωi ) associated with sample point ω i such that -  (i) 0 ≤ P (ωi ) ≤ 1 (ii) ∑P(ωi) for all ωi ∈ S = 1 (iii) P(A) = ∑P(ωi)for all ωi ∈A. The number P (ωi ) is called probability of the outcome ωi .

14. Equally likely outcomes: All outcomes with equal probability

15. Probability of an event: For a finite sample space with equally likely outcomes Probability of an event P(A) = n(A)/n(S) , where n(A) = number of elements in the set A, n(S) = number of elements in the set S.

16. If A and B are any two events, then P(A or B) = P(A) + P(B) – P(A and B) equivalently, P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

17. If A and B are mutually exclusive, then P(A or B) = P(A) + P(B)

18. If A is any event, then P(not A) = 1 – P(A)

Statistics

In this chapter, you have studied the following points:

1. Facts or figures, collected with a definite purpose, are called data.

2. Statistics is the area of study dealing with the presentation, analysis and interpretation of data.

3. How data can be presented graphically in the form of bar graphs, histograms of uniform width, and of varying widths and frequency polygons.

4. The three measures of central tendency for ungrouped data are:

(i) Mean : It is found by adding all the values of the observations and dividing it by the total number of observations. It is denoted by x̅.

(ii) Median : It is the value of the middle-most observation (s). 

If n is an odd number, the median = value of the (n+1)/2 -th term observation.

If n is an even number, median = Mean of the values of the  (n/2)th and (n/2 +1)th observations.

(iii) Mode : The mode is the most frequently occurring observation.

5. Frequency polygons can also be drawn independently without drawing histograms. For this, we require the mid-points of the class-intervals used in the data. These mid-points of the class-intervals are called class-marks. Class-mark = (Upper limit + Lower limit) / 2.

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