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## Function Representing a DistributionConsider the simple frequency distribution:
^{3}.
Recollect, y = x A function that defines the relationship between the two sets of data in a distribution can be identified. ## UseIf we know the definition of the relationship between the two sets of data in a distribution, we can find the second coordinate relevant to a given first coordinate using this definition. Say for x= 50, f(x) or y = 1,25,000, since y = x^{3}
Consider the simple frequency distribution:
## Deriving the relation between any two variablesThe relationship between any two variables can be derived using a set of values relating to the variables. The relationship can be built using a number of statistical and mathematical techniques#### Regression EquationA Linear Relation between the variables can be found out using Regression Equations. Linear relationship between two variables implies that the equation defining the relationship between the two variables is a first degree equation i.e. a straight line equation of the form y = a + bx or ax + by + c = 0.This is called a linear relationship because, it produces a straight line if we plot the corresponding values of the variables on a graph sheet. #### Normal EquationsA linear and parabolic relationship between the variables can be found out using normal equations.A parabolic relationship between two variables implies that the equation defining the relationship between the two variables is a second degree equation i.e. a curve of the form y = a + bx + cx². #### Geometric TechniqueA Linear Relation between the variables can be found out using the analytical geometric formula for building the equation of a straight line passing through two given points. For this we need just two sets of values for the two variables.
## Some Terms to help your understanding Hide/Show
## Relation between the two values in a probability distributionThe two variables in a probability distribution are the value carried by a variable indicating the range of a random variable and its probability.
## Probability FunctionA function which gives the relationship between the two variables in a probability distribution is named the "Probability Function". A probability function assumes the "variable" indicating the values within the range of a random variable as its independent variable and the "probability" as the dependent variable.In a probability function the value of the "variable" (indicating the values within the range of a random variable) is assumed and its respective probability is found out. ## Denoting the Probability FunctionWe denote the "variable" (indicating the values within the range of a random variable) as "x" and the respective probability as P(X=x). Using these the probability function is denoted as
## Probability Mass Function (pmf)A probability function relating to a discrete random variable is called the "Probability Mass Function".If the range of a random variable "X" assumes a discrete set of values The pmf assigns a probability [P(X = x It gives the probability that the variable (representing the range of the discrete random variable) equals to some value. A discrete probability function, f(x), is a function that satisfies the following properties. **f (x**_{i}) ≥ 0 The probability for the variable to carry a particular value is always a positive real number.**Σ f (x**_{i}) = 1 [i = 1, 2, 3, ... ∞] The sum of the probabilities of all the possible values that the variable (representing the range of the discrete radom variable) may carry is equal to One.
Considering the probabiliity distribution as above
For each value that the variable may carry the probabilty is equal to 1/6. [Such a function where the dependent variable carries the same value for all the values that the independent variable may carry is called a constant function] |

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