Random Walk Probability Distribution

By Author: Pierce Brosnan
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Introduction for random walk probability distribution:

In random walk probability distribution, consider a random testing whose sample space is S. A random walk X is a function from the sample space S into the set of real numbers `RR` such that for each interval I in `RR`, the set {s `in` S |X(s) `in` I} is an event in S.

In a particular experiment, a random walk X would be some function that assigns a real number X(s) to each possible outcome s in the sample space. Given a random experiment, there can be many random walks.

Two types of random walk:

If the space of random walk X is countable, then X is called a discrete random walk.

If the space of random walk X is uncountable, then X is called a continuous random walk.

Discrete Random Walk for Random Walk Probability Distribution:

In random walk probability distribution, let `R_X` be the space of the random walk X. The function f : `R_X` `->` `RR` defined by

f(x) = P(X = x)

is called the probability density function (pdf) of X.

Example Discrete random walk for random walk probability distribution:

In ...
... random walk probability distribution, an introductory statistics class of 50 students, there are 11 freshman, 19 sophomores, 14 juniors and 6 seniors. One student is selected at random. What is the sample space of this experiment? Construct a random walk X for this sample space and then find its space. Further, find the probability density function of this random walk X.

Solution:

The sample space of this random experiment is

S = {Fr, So, Jr, Sr}.

Define a function X : S `->` `RR` as follows:

X(Fr) = 1, X(So) = 2

X(Jr) = 3, X(Sr) = 4.

Then clearly, X is a random walk in S. The space of X is given by

RX = {1, 2, 3, 4}.

The probability density function of X is given by

f(1) = P(X = 1) = `11/50`

f(2) = P(X = 2) = `19/50`

f(3) = P(X = 3) = `14/50`

f(4) = P(X = 4) = `6/50`

Continuous Random Walk for Random Walk Probability Distribution:

In random walk probability distribution, let X be a continuous random walk whose space is the set of real numbers `RR`. A nonnegative real valued function f : `RR` `->` `RR` is said to be the probability density function for the continuous random walk X. if it satisfies:

(a) `int_(-oo)^(oo)`f(x) dx `=` 1, and

(b) If A is an event, then P(A) `=` `int_A`f(x) dx.

Example for Continuous random walk for random walk probability distribution:

In random walk probability distribution, is the real valued function f : `RR` `->` `RR` defined by

f(x) = 2`x^(-2)` if 1 `