Unit – III Spatial data Ajustment.pdf

1. UNIT – III VARIANCE COVARIANCE PROPAGATION

2. VARIANCE COVARIANCE PROPAGATION  Random events and probability  Random variables  continuous probability distributions  normal distribution  Expectation  measures of precision and accuracy  covariance and correlation  covariance, cofactor and weight matrices  Introduction to sampling  Derivation of the propagation laws - Examples  stepwise propagation

3. Random Variables • In an experiment, a measurement is usually denoted by a variable such as X. • In a random experiment, a variable whose measured value can change (from one replicate of the experiment to another) is referred to as a random variable. •A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes in a sample space to a measurable space, often the real numbers.

4. Random Variables

5. Random Variables

7. Probability • Used to quantify likelihood or chance • Used to represent risk or uncertainty in engineering applications •Can be interpreted as our degree of belief or relative frequency •Probability statements describe the likelihood that particular values occur. • The likelihood is quantified by assigning a number from the interval [0, 1] to the set of values (or a percentage from 0 to 100%). • Higher numbers indicate that the set of values is more likely.

8. Probability • A probability is usually expressed in terms of a random variable. • For the part length example, X denotes the part length and the probability statement can be written in either of the following forms • Both equations state that the probability that the random variable X assumes a value in [10.8, 11.2] is 0.25.

9. Probability Complement of an Event • Given a set E, the complement of E is the set of elements that are not in E. The complement is denoted as E’. Mutually Exclusive Events • The sets E1 , E2 ,...,Ek are mutually exclusive if the intersection of any pair is empty. That is, each element is in one and only one of the sets E1 , E2 ,...,Ek . Probability Properties

11. Probability of an Event E. Suppose that the sample space S = {o1, o2, o3, … oN} has a finite number, N, of outcomes. Also each of the outcomes is equally likely (because of symmetry). Then for any event E   ( ) ( ) ( ) no. of outcomes in = total no. of outcomes n E n E E P E n S N = =

19. Discrete Random Variables Probability Mass Function

20. Discrete Random Variables Cumulative Distribution Function

21. Discrete Random Variables Cumulative Distribution Function

23. p(x) is Probability function

25. F(x) is Probability Distribution function

36. F(x) is Probability Distribution function f(x) is Probability Density function

38. Continuous Random Variables Probability Density Function • The probability distribution or simply distribution of a random variable X is a description of the set of the probabilities associated with the possible values for X.

39. Continuous Random Variables Probability Density Function

40. Continuous Random Variables Probability Density Function

49. Normal Distribution Undoubtedly, the most widely used model for the distribution of a random variable is a normal distribution.

50. Normal Distribution

68. Random Samples, Statistics, and The Central Limit Theorem

73. EXPECTATION

74. EXPECTATION

75. EXPECTATION

76. Expectation for Continuous Probability Distribution

81. Measures of Precision and Accuracy

86. Covariance and Correlation Covariance and correlation are two terms that are exactly opposite to each other. However, they both are used in statistics and regression analysis. Covariance shows us how the two variables vary, whereas correlation shows us the relationship and how they are related. Correlation and covariance are two statistical concepts used to determine the relationship between two random variables. Correlation defines how a change in one variable will impact the other, while covariance defines how two items vary together.

87. Covariance and Correlation

94. Covariance, Cofactor and Weight Matrices

105. Introduction to sampling

106. Propagation laws

119. Propagation

120. Propagation

121. Propagation

122. Stepwise Propagation

Unit – III Spatial data Ajustment.pdf

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Unit – III Spatial data Ajustment.pdf