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Probability Mass Function of a Discrete Random (Stochastic or Chance) Variable

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... From Page 3
   
 

Function Representing a Distribution

Consider the simple frequency distribution:
Variable (x) 5 8 10 15 50
Frequency (y) 125 512 1,000 3,375 8,000
The two variables in the distribution are related by the relation y = x3.

Recollect, y = x3 is a function. It can also be written as f (x) = x3.

A function that defines the relationship between the two sets of data in a distribution can be identified.

Use

If 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 = x3 Consider the simple frequency distribution:
x 5 10 15 20
y
15
129
30
129
45
129
60
129
The two variables in the distribution are related by the relation 129y − 3x = 0 (Or) y =
3x
129

Deriving the relation between any two variables

The 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 Equation

    A 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 Equations

    A 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 Technique

    A 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

In a function defining the reltionship between two variables one of the variable represents the values within the domain and the other the values within the range.
  • Dependent and Independent Variables

    • Independent Variable

      The variable representing the values within the domain is called the independent variable.

      This value is assumed and the related value within the range of the function is found out always.

    • Dependent Variable

      The variable representing the values within the range is called the dependent variable.

      This value is found out based on the value of the independent variable. Since its value is dependent on what value we have chosen for the independent variable it is called a dependent variable.

    Eg: 1. y = 3 + 4x is a function from "x" to "y". It defines the relationship between the two variable "x" and "y".

    Using y = 3 + 4x, we assume a value for "x" and find the related value for "y".

    "x" is called the independent variable and "y" the dependent variable.

    2. f(x) = 4x² − 24; "f" is a function from "x" to "__". It defines the relationship between the two variable "x" and another variabel which we generally name "y".

    f(x) indicates that there is another variable which is dependent on "x" or which is a function of "x"

    "x" is called the independent variable and the other variable say "y" the dependent variable.

  • Implicit and Explilcit Functions

    • Explicit Function

      A function in which the dependent and independent variables are clearly identifiable is called an explicit function.

      It shows the dependent variable expressed in terms of the independent variable.

      y = a = bx → "y" the dependent variable is expressed in terms of "x" the independent variable.

    • Implicit Function

      A function in which the dependent and independent variables are not clearly identifiable is called an implicit function.

      3x + 4y = 24.

      This is a function since it defines the relationship between the two variables "x" and "y. It would not be possible for us to say which is the dependent and which is the independent variable in this.

      Say if we write it as y =
      24 − 3x
      4

      We can say "x" is the independent variable and "y" the dependent variable.

      We can also write it as x =
      24 − 4y
      3

      We can say "y" is the independent variable and "x" the dependent variable

If an equation is to be called a function, then it can be so only if it has a domain and a range i.e. only when its independent and dependent variables are clearly identified.

Therefore, the same equation cannot be said to be a function from the "First Variable" to the "Second Variable" as well as a function from the "Second Variable" to the "First Variable". However, as in the above case of an implicit function, it would not be possible to define which is the dependent and which is the independent variable unless it is specified.

Relation between the two values in a probability distribution

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A distribution indicating all the possible numerical values (within the range of a discrete random variable in relation to an experiment) with their respective probabilities is called a discrete probability distribution.

The two variables in a probability distribution are the value carried by a variable indicating the range of a random variable and its probability.
Eg: 1. The probability distribution of a random variable relating to the experiment of tossing 3 coins where getting a head is termed a success.
x 0 1 2 3
P(X=x)
1
8
3
8
3
8
1
8

Probability Function

A 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 Function

We 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

      "f" is a function from "x" to P(X=x) (Or) f : "x" → P(X=x) (Or) f(x) = P(X=x)

Eg: 1. The probability distribution of a random variable relating to the experiment of tossing 3 coins where getting a head is termed a success and "x" is a variable indicating the number of successes.
x 0 1 2 3
P(X=x)
1
8
3
8
3
8
1
8
Where, f(x) = P(X=x) represents the probability function relating to this probability distribution

Where "x =0", f(0) = P(X=0); Where "x = 2" f(2) = P(X=2); ...

2. A person on tossing a die wins an amount equal to the "number on the die × Rs. 10" if an even number appears and loses an amount equal to the "number on the die × Rs. 5" if an odd number appears.

In the experiment of tossing a coin,

The Sample Space "S" = {ONE, TWO, THREE, FOUR, FIVE, SIX}

If "s' represents the elements of the set "S" and "x" the amount that the person wins,

Then, the random variable "X" representing the relationship between "s" and "x" is

X(s) = x (Or) X : s → x

The random variable distribution would be
Number on the die ONE TWO THREE FOUR FIVE SIX
x − 5 + 20 − 15 + 40 − 25 + 60

"X" is a function with domain "S" and Range = {1, 2, 3, 4, 5, 6}.

The probability distribution indicating the values that "x" (representing the values within the range of "X") may carry with the respective probabilities would be
x − 5 + 20 − 15 + 40 − 25 + 60
P(X=x)
1
6
1
6
1
6
1
6
1
6
1
6
Where, f(x) = P(X=x) represents the probability function relating to this probability distribution

Where, "x = −5", f(−5) = P(X=−5); Where "x = +10", f(+10) = P(X=+10); ...

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 x1, x2, x3, ... xn, then the function "f" defined by f(xi) = P(X = xi) is called the "Probability Function" or "Probability Mass Function" of "X".

The pmf assigns a probability [P(X = xi)] for each of the possible values [xi] of the variable.

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 (xi) ≥ 0
    The probability for the variable to carry a particular value is always a positive real number.
  • Σ f (xi) = 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.
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Considering the probabiliity distribution as above

x − 5 + 20 − 15 + 40 − 25 + 60
P(X=x)
1
6
1
6
1
6
1
6
1
6
1
6
Here, f(x) = P(X=x) ⇒ f(x) =
1
6

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]

Author Credit : The Edifier ... Continued Page 5
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