subject: Random Walk Probability Distribution [print this page] 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
