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Sample space is the set of all possible outcomes of a random experiment; it may be discrete or continuous. Events are subsets of the sample space, and they represent the questions or conditions we are interested in within that space. A probability measure assigns a numerical value to each event (satisfying the axioms of probability). A random variable is a measurable function from the sample space to the real numbers; it does not arise directly from the probability measure, but is defined on the probability space. The probability distribution is the induced measure of a random variable on the real line (or another target space). It describes how probability mass or density is allocated over the values that the random variable can take. Example we have 2 dice we wanna through so
  • sample space will be 6x6 = 36 results = {(1,1),(1,2),...,(6,6)}
  • Event will be a question like (outcome has sum = 7) whici is a subset of the space = {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}
  • probability measure gives a number to each event so for above Event (sum =7) call it A > P(A) = 6/36 = 1/6
  • Random Variable X: Ī© ā†’ ā„ (let say sum of both dice) X(1,1) = 2 X(1,2) = 3 X(2,3) = 5 X(3,4) = 7 X(6,6) = 12 ā€¦
  • Probability Distribution
    X (Sum)Possible OutcomesCount( P(X = x) )
    2(1,1)11/36
    3(1,2), (2,1)22/36
    4(1,3), (2,2), (3,1)33/36
    5(1,4), (2,3), (3,2), (4,1)44/36
    6(1,5), (2,4), (3,3), (4,2), (5,1)55/36
    7(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)66/36 ā† Highest
    8(2,6), (3,5), (4,4), (5,3), (6,2)55/36
    9(3,6), (4,5), (5,4), (6,3)44/36
    10(4,6), (5,5), (6,4)33/36
    11(5,6), (6,5)22/36
    12(6,6)11/36
in abstract
  • we can say random variable is a representation of group of events under specific category let say we have events A (sum of nerds = 7) B (sum of nerds = 8) ā€¦ so many so we can categorize them all under specific category which is sum and here we could have a random variable X(sum of nerds) so its a mathematical representation of uncertinity Random variables allow us to represent many related events under one mathematical framework, but not every event naturally fits into a single random variable so outcome is insteaed of writiing 10s of events merge them all in one mathematical condition