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Random Experiment: A random experiment is a process by which we observe something uncertain. After the experiment, the result of the random experiment is known. An outcome is a result of a random experiment. Example: Throwing a die, a pack of cards. Trial: Any particular performance of a random experiment is called a trial Sample space: The set of all possible outcomes of a statistical experiment is called the sample space. An event is a subset of a sample space. Example: roll a die; sample space: S={1,2,3,4,5,6}. If we would like to know the probability of getting even number, then event is {2,4,6}.
Statistical Definition:
If a random experiment or trail results in ‘ ’ possible outcomes(or cases), out of which ‘ ’ are favourable to the occurrence of an event , then the probability ‘p’ of occurrence of , denoted by ( ), is given by:
(i) Since ≥ 0, ≥ 0 ≤ ≥ 0 ≤ 1 0 ≤ ≤ 1
(ii) + = 1
Bayes Theorem of Probability:
Random Variable
Let S be the sample space associated with a given random experiment E. A real-valued function defined on S and range is R (−∞, ∞), called a one-dimensional random variable. If the function values are ordered pairs of real numbers (i.e., vectors in two- space), the function is said to be a two-dimensional random variable. More generally, an n-dimensional random variable is simply a function whose domain is S and whose range is a collection of n-tuples of real numbers (vectors in n-space).
Definition: A random variable is a function that associates a real number with each element in the sample space. Note: One-dimensional r.v. will be denoted by capital letters, X, Y, Z….etc. The values which X, Y, Z….etc, can assume are denoted by lower case letters., x, y, z…etc. Example: Two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. The possible outcomes and the values y of the random variable: Y, where Y is the number of red balls, are
:
Example: Answer:
Continuous Random Variables
A random variable X is said to be a continuous random variable if it takes all possible values between certain limits or in an interval which may be finite or infinite. Example: operating time between two failures of a computer Probability density function (p.d.f):
Distribution Function (or) Cumulative Distribution Function (cdf) Relationship between cumulative distribution function and probability distribution function
Problem 1: If the random variable takes the values 1,2,3 and 4 such that 2 P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4) , find the probability distribution and the cumulative distribution function of X.
Problem 3:
