STAT1201 - Lecture 2, week 3 3.2 - Randomness and Probability Theory p2 Practice - Difference between discrete and continuous random variables - Discrete and continuous probability distributions - Expected value and standard deviations of discrete probability distributions Random Variables A random variable is a random process with numerical outcomes e.g., the number of text messages students receive during this lecture, possible outcomes 0,1,2, N ... Discrete Random Variable - A random variable that has a countable number of possible values e.g., number of children in a family or number of left-handed students in STAT1201; things that are counted not measured Continuous Random Variable - A random variable where the data can take infinitely many values e.g., heights of the STAT1201 students or blood haemoglobin level; things that are measured not counted Discrete Probability Distribution - The listing of all possible values of a discrete random variable X along with their associated probabilities e.g., Define X = number shown by rolling a six-sided dice (X = 1, 2, 3, 4, 5, 6.) then the probability distribution of X can be written as follows P (X)= 1 (1/6), 2 (1/6), 3 (1/6), 4 (1/6), 5 (1/6), 6 (1/6)
STAT1201 - Lecture 2, week 3 Example - The following table shows the probability distribution of the number of children (X) in a family and the associated probabilities from a random sample of families living in Brisbane | X| P (X=x) 1 --: | :---- 1 1 0|0.21 1 1|0.45 1 1 2|0.23 I 3|0.11 What is the probability that no more than two children are in a family? P(X<2)=P(X=0)+P(X=1)+P(X=2) P(X<2)=0.89 The most popular discrete probability distributions are Binomial Distribution and Poisson Distribution The most popular continuous probability distributions are Uniform Distribution, Normal Distribution, Exponential Distribution, T-Distribution, Chi-Squared Distribution and F- Distribution The expected value of Mean or (E(X)) - Long-run average of a random variable. If we repeat taking random samples of families living in Brisbane, the mean or expected number of children can be found as follows E(X)=1=20. 1(I = 0) Using the children's distribution E(X) =[]=0x0.21+ 1x0.45 + 2x0.23 + 3x0.11 = 1.24 The variance of (VAR(X)) We can quantify the variability of a discrete random variable using squared deviations about the mean as we did for a sample of data Var(X) = 20(0 = 1)(1- 1) 2 SD(X)=100(0)
STAT1201 - Lecture 2, week 3 Continuous Probability Distribution - Functions cannot be presented in a table or histogram as we did for discrete probability distributions as there are an uncountable number of possible outcomes The probability of an individual outcome is 0 P (X)=0 We always calculate the probability for a range of continuous random variables X ? We can use the concept of integral to calculate these probabilities. E(X)=1=100-00 1(0)111 Var(X) = 700-00 1 (0) (-1)21? Some important rules to know Suppose X is a random variable Let Y = aX where a is a constant Then E (Y) = aE(X) and Var(Y) = [] 2Var(X) Let Y =