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4. Random Variables A random variable is a way of recording a quantitative variable of a random experiment. 4. Random Variables A random variable is a way of recording a quantitative variable of a random experiment. In particular Chapter 4 talks about discrete random variables. 4. Random Variables A random variable is a way of recording a quantitative variable of a random experiment. In particular Chapter 4 talks about discrete random variables. If a random variable has a particular distribution (such as a binomial distribution) then our work becomes easier. We use formulas and tables. 5. Continuous Random Variables A random variable is a way of recording a quantitative variable of a random experiment. In particular Chapter 4 talks about discrete random variables. If a random variable has a particular distribution (such as a binomial distribution) then our work becomes easier. We use formulas and tables. 5. Continuous Random Variables A random variable where X can take on a range of values, not just particular ones. Examples: Heights Distance a golfer hits the ball with their driver Time to run 100 meters Electricity usage of a home. Continuous probability distribution functions For a discrete random variable, probabilities are given as a table of values, and the distribution can be graphed as a bar graph. For a continuous random variable, probabilities are specified by a continuous function. The graph of the probability distribution function is a curve. Figure 5.1 A probability f(x) for a continuous random variable x Copyright © 2013 Pearson Education, Inc.. Definition Copyright © 2013 Pearson Education, Inc.. Figure 5.2 Density Function for Friction Coefficient, Example 5.1 Copyright © 2013 Pearson Education, Inc.. Find probability friction is less than 10. Copyright © 2013 Pearson Education, Inc.. Find probability friction is less than 10. Solution:Probabiity = area of shaded triangle = (1/2)(5)(0.2)=0.5 Copyright © 2013 Pearson Education, Inc.. Uniform Distribution A Uniform Distribution has equally likely values over the range of possible outcomes. Uniform Distribution A Uniform Distribution has equally likely values over the range of possible outcomes. A graph of the uniform probability distribution is a rectangle with area equal to 1. Example The figure below depicts the probability distribution for temperatures in a manufacturing process. The temperatures are controlled so that they range between 0 and 5 degrees Celsius, and every possible temperature is equally likely. P(x) 0.2 0 0 1 2 3 4 Temperature (degrees Celsius) 5 x Example Note that the total area under the “curve” is 1. P(x) 0.2 0 0 1 2 3 4 Temperature (degrees Celsius) 5 x Example P(x) 0.2 0 0 1 2 3 4 5 x Temperature (degrees Celsius) What is the Probability that the temperature is exactly 4 degrees? Example P(x) 0.2 0 0 1 2 3 4 5 x Temperature (degrees Celsius) What is the Probability that the temperature is exactly 4 degrees? Answer: 0 Explanation Since we have a continuous random variable there are an infinite number of possible outcomes between 0 and 5, the probability of one number out of an infinite set of numbers is 0. Example What is the probability the temperature is between 10C and 40C? P(x) 0.2 0 0 1 2 3 4 Temperature (degrees Celsius) 5 x Example What is the probability the temperature is between 10C and 40C? P(x) 0.2 0 0 1 2 3 4 Temperature (degrees Celsius) 5 x What is the probability the temperature is between 10C and 40C? P(x) 0.2 0 0 1 2 3 4 5 x Temperature (degrees Celsius) We know that the total area of the rectangle is 1, and we can see that the part of the rectangle between 1 and 4 is 3/5 of the total, so P(1 x 4) = 3/5*(1) = 0.6. Review: Probabilities and Area For a density curve depicting the probability distribution of a continuous random variable, – the total area under the curve is 1, – there is a direct correspondence between area and probability. – Only the probability of an event occurring in some interval can be evaluated. – The probability that a continuous random variable takes on any particular value is zero. General Uniform Distribution A Uniform Distribution has equally likely values over the range of possible outcomes, say c to d. 1 Height of the density function : f(x) d c cd Mean 2 d c Standard Deviation 12 Normal Distributions This is the most common observed distribution of continuous random variables. A normal distribution corresponds to bellshaped curves. Normal Distributions This is the most common observed distribution of continuous random variables. A normal distribution corresponds to bellshaped curves. y e ( x ) 2 / 2 2 2 Reminder: Mu is the mean, sigma is the standard deviation. Examples The following are examples of normally distributed everyday data. – Grades on a test. – How many chips are in a small bag of potatoe chips. – The measurements of distance between two points. – The heights of students in this class. Normal Distributions Normal Distributions Shape of this curve is determined by µ and σ – µ it’s centered, σ is how far it’s spread out. Standard Normal Distribution The Standard Normal Distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1. 0, 1 In this way the formula giving the heights of the normal curve is simplified greatly. Z-score Standard Normal Probabilities P(0 z 1) represents the probability that z takes on values between 0 and 1, which is represented by the area under the curve between 0 and 1. P(0 z 1) = 0.3413 P(0 z 1) = 0.341 Revelation! Since the mean is 0 and the standard deviation is 1, this tells us that the probability that z is within one standard deviation of the mean (either below or above) is (2)(0.341)= 0.682. P(0 z 1) = 0.341 Revelation! Since the mean is 0 and the standard deviation is 1, this tells us that the probabiity that z is within one standard deviation of the mean (either below or above) is (2)(0.341)= 0.682. Agrees with Empirical Rule: 68% of data lies within one standard deviation of the mean Finding Probabilities when given z-scores. For a given z-score, the probability can be found in a table in the back of the text (Table IV), also see inside front cover. Note: The table only gives the areas under the curve to the right between 0 and z. To find other intervals requires some tricks. Table 5.1 Copyright © 2013 Pearson Education, Inc.. Find probability z is between 1.33 and +1.33. Copyright © 2013 Pearson Education, Inc.. Want probability z is between -1.33 and +1.33. Solution: Locate 1.33 in the row labeled 1.3 and the column labeled .03. By symmetry, ans = 2(0.4082) = .8164 Copyright © 2013 Pearson Education, Inc.. Find probability z exceeds 1.96 in absolute value. Copyright © 2013 Pearson Education, Inc.. Areas under the standard normal curve for z exceeding 1.96 in absolute value Copyright © 2013 Pearson Education, Inc.. Areas under the standard normal curve for z exceeding 1.96 in absolute value Revelation! It follows that the area of the unshaded region is 0.95. Agrees with Empirical Rule which states that, for data sets having a mound shaped distribution, 95% of the values lie within approximately 2 standard deviations of the mean Keys to success Learn the standard normal table and how to use it. We will be using these tables through out the course.