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Introduction and 7.1 Discrete and Continuous Random Variables Introduction Random Variable A Random Variable is a variable whose value is a numerical outcome of a random phenomenon. In this section, we will learn two ways of assigning probabilities to events. These two models will dominate our application to probability to statistical interference. D Discrete Random Variables A Discrete Random Variables X has a countable number of possible values. The Probability distribution of X lists the values and their probabilities. Value X1 X2 X3 … Xk of X Probabi P1 P2 P3 … Pk lity Example 7.1 Getting Good Grades An instructor of a large class assigns grades in the following manner. What is the Probability that a student gets a B or better? Grade 0 1 2 3 4 (F) (D) (C) (B) (A) Probab 0.10 0.15 0.30 0.30 0.15 ility Solution P(grade is 3 or 4) = P(X = 3) + P(X=4)= 0.3+0.15= 0.45 Probability Histograms Random Digits 2 0 1 2 3 4 5 6 7 8 9 Random Digits Benford’s Law 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 Benford’s law Example 7.2 Tossing Coins What is the probability distribution of the discrete random variable X that counts the number of heads in four tosses of a coin? HTTH HTHT HTTT THTH HHHT THTT HHTT HHTH TTHT THHT HTHH TTTT TTTH TTHH THHH HHHH X=0 X=1 X=2 X=3 X=4 Probability of each value of X P(X=0)=1/16=0.0625 P(X=1)= 4/16=0.25 P(X=2)= 6/16=0.37 P(x=3)=4/16= 0.25 P(X=4) =1/16=0.0625 # of heads 0 1 2 3 4 Probability 0.625 0.25 0.37 0.25 0.625 Continuous Random Variables Choosing a random # between 0 and 1. S={all numbers x such that 0≤x≤1} We cannot assign probabilities to each individual value of x and then sum, because there are infinitely many possible values. Instead we use areas under a density curve to assign probability to events. Any density curve has an area of exactly 1 underneath corresponding to a probability of one. Example 7.3 Random Numbers and the Uniform Distribution P(0.3≤X≤0.7) 1 P(X≤0.5 or X>0.8) =0.5 +0.2= 0.7 The probability distribution of a continuous random variable assigns probabilities as area under a density curve. Continuous Random Variable A Continuous Random Variable X takes all values in an interval of numbers. The Probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event. All continuous probability distributions assign probability 0 to every individual outcome. We can ignore the distinction between > and ≥ when finding probabilities for continuous(but not discreet) random variables. Normal Distributions as Probability Distributions Because any density curve describes an assignment of probabilities, normal distributions are probability distributions. Recall that N(µ,σ) stands for a normal distribution with mean µ and standard deviation σ. If X has the N(µ,σ) distribution, then the standardized variable Z=X-µ σ is a standardized normal random variable having the distribution N(0,1). Example 7.4 Drugs in Schools N(0.3,0.0118) What is the probability that the poll results differ from the truth about the population by more than 2 percentage points. Assignment P. 401-404 7.6, 7.10, 7.13, 7.15