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Understanding Discrete Random Variables and Binomial Probability in Statistics

This lecture focuses on discrete random variables, specifically exploring the binomial probability distribution. It introduces the concept of a discrete random variable, explaining its characteristics and how it can take on integer values. The lecture covers the probability function, expected value, and variance, using practical examples like tossing coins and family planning scenarios. It distinguishes between sampling with and without replacement and discusses the conditions for binomial distribution to hold true. Key mathematical formulas and real-life applications are also included.

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Understanding Discrete Random Variables and Binomial Probability in Statistics

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  1. Stat 13 Lecture 19 discrete random variables, binomial • A random variable is discrete if it takes values that have gaps : most often, integers • Probability function gives P (X=x) for every value that X may take • X= number of heads in 3 tosses • A couple decides to have children till at least one of each sex or a maximum of 3; X=number of girls. (tree)

  2. Expected value and standard deviation • E(X) = sum of P(X=x) x (weighted average; using probability as weight) • Var (X) = sum of P(X=x) (x- E(X))2

  3. Binomial probability • Coin tossing ; multiple choices ; formula of binomial ; combination number • P(X=x)= ( ) px (1-p)(n-x) • Sampling with replacement • Sampling without replacement; infinite population • Sampling without replacement, finite population; (opinion ) survey sampling n x

  4. Conditions for binomial to hold • Model the number of successful trials out of n trials • Must know n • Must know (or be able to estimate) p (=prob of success in each trial) • Must satisfy independence assumption in different trials • p should be the same in each trial

  5. Sampling without replacement • Suppose in a population of N individuals, a random sample of n individuals are selected. Their opinions on a proposal are recorded. Suppose in the population the proportion of individuals saying yes is p. Then X, the number of individuals in the sample saying yes follows a hypergeomtric distribution • P(X=x)= [Np choose x][N(1-p) choose (n-x)]/ [N choose n], which is approximately equal to binomial when N is large and the sampling fraction n/N is small.

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