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Unit II: Summarizing Data & Revisiting Probability (NOS 2101) 2.3.Probability: Probability is the chance of occurrence anything. P(A)= S/P Where S is sample size or no of positive outcomes and P is the population size or total no of outcomes. Probability distribution : Which describes how the values of a random variable are distributed Binomial distribution: The collection of all possible outcomes of a sequence of coin tossing Normal distribution: The means of sufficiently large samples of a data population Note: The characteristics of these theoretical distributions are well understood, they can be used to make Statistical inferences on the entire data population as a whole. Example: Probability of ace of Diamond in a pack of 52 cards when 1 card is pulled out at random. “At Random” means that there is no biased treatment No. of Ace of Diamond in a pack = S = 1 Total no of possible outcomes = Total no. of cards in pack = 52 Probability of positive outcome = S/P = 1/52 That is we have 1.92% chance that we will get positive outcome. 2.4.Expected value: The expected value of a random variable is the long-run average value of repetitions of the experiment it represents. Example: The expected value of a dice roll is 3.5 means the average of an extremely large number of dice rolls is practically always nearly equal to 3.5. Expected value is also known as the expectation, mathematical expectation, EV, mean, or first moment. • Expected value of a discrete random variable is the probability-weighted average of all possible values • Continuous random variables is the sum replaced by an integral and the probabilities by probability densities. 2.5.Random Variable: A random variable, aleatory variable or stochastic variable is a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense). A random variable can take on a set of possible different values (similarly to other mathematical variables), each with an associated probability, in contrast to other mathematical variables. A random variable is a real-valued function defined on the points of a sample space. Page | 1 Random variables are two broad categories: Random variable with discrete values Bivariate Random Variable Discrete: specified finite or countable list of values, endowed with a probability mass function, characteristic of a probability distribution; Continuous: Any numerical value in an interval or collection of intervals, via a probability density function that is characteristic of a probability distribution; Mixture of both types: The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates. Example: If we toss a coin for 10 times and we get heads 8 times then we cannot say that the 11th time if coin is tossed then we get a head or a tail. But we are sure that we will either get a head or a tail. 2.6.Bivariate Random Variable: Bivariate Random Variables are those variables having only 2 possible outcomes. Example: Flip of coin(two outcomes: head/tail). Page | 2