Chapter 4 • Probability distributions deal with the distributions of characteristics of a given population. Distribution describes the pattern on how the values of a variable vary. • Discrete Random Variable: A variable can be discrete, such as # of accidents occurred at an intersection per month, or • Continuous Random Variable: The possible values are in a real interval, such as SAT score (between 0 to 1600) , gas price (between $0 to $10 per gallon). • In these two chapters, we will focus on discrete variable and introduce a very useful special discrete distribution: Binomial distribution. General Objectives: 1. What is a random variable and the types of random variables? 2. What is a discrete random variable and its distribution? 3. What is a Binomial distribution and how it is used in real world applications?A variable X is a random variable if the value that it assumes corresponding to the outcome of an experiment, is a chance or random event. NOTE: In Chapter 15, the outcomes can be anything such as Head, Tail, Win/Loss, and so on. In this chapter, similar process is applied to develop the probability distribution, except that the possible outcomes will be numerical values. [The random variables introduced in this chapter are special cases of those in Chapter 15] Two types of random variables: Discrete – The possible values of the random variable is finite or countable. Example: X = # of games UCLA will win in a football season. X = # of defective parts in inspecting a random sample of 10 parts. X = # of hijackings per year X = # of school violence per month in USA Continuous – The possible values are in a real interval. Example: X = SAT scores, X = adult height, X = Quarterly profit of a company. X = The damage of house fire. 23 Discrete Random Variable and Probability Distribution The probability distribution for a discrete random variable is a formula, table, or graph that provides p(x), the probability associated with each of the values of x. Requirements for a Discrete Probability Distribution [ A Must Know property] 1)(0  xp 1)(xpA typical tabular form of a discrete distribution is X x1 x2 ……… xn P(X) P(x1) P(x2) ……… P(xn) A simple example of a discrete random variable: # of heads occur when tossing two fair coins together: X: # of Heads 0 1 2 P(X) 1/4 1/2 1/4 WHY?Exercise: Are the following discrete distributions valid? [Similar exam questions] x -2 0 3 7 P(x) .4 .2 .3 .2 x -2 0 3 7 P(x) -.2 .5 .4 .3 x -10 -4 3 8 P(x) .2 .35 .3 .15 Graphical presentation of a discrete probability distribution is : Probability Bar Chart, with P(x) on Y-axis and X-values on X-axis. P(x) X 1) 2) 3) 45 Activity for Discrete Distribution [Similar exam questions] Insurance companies decide their premium based on the # of accidents and the type of accidents occurred in the previous years in a large area. The following is an example about # of car accidents per day in a large city area: X : # of Accidents: 0 1 2 3 4 5 6 Probability .05 ? .3 .2 .1 .08 .02 Q1: Determine P(X =1 ) Q2: Draw a bar graph of the probability distribution of # of a accidents. Q3: What is the probability of having exactly 3 accidents in a given day? Q4: What is the probability of having at most 2 accidents in a given day? Q5: What is the probability of having at least 3 accidents in a given day? Q6: What would the expected average number of accidents in a given day? Q7: What would the variance of the number of accidents in a given day? Q8: What is the standard deviation of the number of accidents in a given day?6 In order determine the premium, the insurance company would ask: 1) What is the expected average # of accidents? 2) Is the # of accidents the same from day to day? Or it varies from day to day? 3) If it varies from day to day, then, what is the variance and standard deviation of the # of accidents? 4) What does the distribution of the # of accidents look like? Expected average # of accidents is similar to the average we learned in Chapters 3 and 4. Is the following a right way to find the expected average # of accidents = (0+1+2+3+4+5+6)/7 = 0 * (1/7) + 1*(1/7) + ….. + 6*(1/7) ? No, this is NOT correct. Each value has different probability to occur. Therefore, we need to take the probability into account: The CORRECT ANSWER: Expected # of accidents = 0*P(0) + 1*P(1) + 2*P(2) + 3*P(3) + 4*P(4) + 5*P(5) + 6*P(6) = 0 (.05) + 1(.25) + 2(.3) +3(.2) +4(.1) + 5(.08) + 6(.02) = 2.37 On average, the expected # of accidents per day is 2.37 in the city.7 How to describe a discrete distribution? Distribution is the pattern that describes the center, the variation, the probability that each possible value occurs and the shape of the distribution [These are must know properties] The probability of each value of X and the shape of graphical presentation: This is best described by a Bar graph. This is the Bar Graph for the Car accident example: NOTE: Shapes of the distribution can be skewed-to-right, symmetric, and skewed-to-left. For this car accident example, the probability distribution is skewed-to-right. AccidentsP(X)65432100.300.250.200.150.100.050.00Probability Bar Graph for the # of AccidentsHow to describe a discrete distribution? The distribution has the center value (or called mean, or expected value) and the variation of the distribution can be described by a numerical value, called variance or by standard deviation. The Center: The mean or expected value of the variable X : m  E(x)  S xP(x) NOTE: The Expected Value m, the population mean. For a given population, there is ONLY one Expected value, m. DO NOT mix up with the Sample Mean, . We can take many different samples from the same population. Sample means are different from sample to sample. BUT, there is only one population mean. You can not compute population mean from a random sample that is not the entire population. The Variation: The variance and standard deviation of X: Let x be a discrete random variable with probability distribution p(x) and mean m. The variance of x is s 2  E [(x - m ) 2 ]  S(x - m ) 2 p(x) = The standard deviation s of a random variable X : s = NOTE: s 2 and s are population parameters. DO NOT mix up with Sample variance, s2 , and sample s.d., s. 22()x P xm-x2s8Exercise Activities for Discrete