Continuous Random Variables & The Normal Probability Distribution

1. Continuous Random Variables & The Normal ProbabilityDistribution

2. Learning Objectives • Understand characteristics about continuous random variables and probability distributions • Understand the uniform probability distribution • Graph a normal curve • State the properties of a normal curve • Understand the role of area in the normal density function • Understand the relation between a normal random variable and a standard normal random variable

3. Continuous Random Variable & Continuous Probability Distribution

4. Continuous Random Variable • The outcomes of a continuous random variable consist of all possible values made up an interval of a real number line. • In other words, there are infinite number of possible outcomes for a continuous random variable.

5. Continuous Random Variable • For instance, the birth weight of a randomly selected baby. The outcomes are between 1000 and 5000 grams with all 1-gram intervals of weight between1000 and 5000 grams equally likely. • The probability that an observed baby’s weight is exactly 3250.326144 grams is almost zero. This is because there may be one way to observe 3250.326144, but there are infinite number of possible values between 1000 and 5000. According to the classical probability approach, the probability is found by dividing the number of ways an event can occur by the total number of possibilities. So, we get a very small probability almost zero.

6. Continuous Random Variable • To resolve this problem, we compute probabilities of continuous random variables over an interval of values. For instance, instead of getting exactly weight of 3250.326144 grams we may compute the probability that a selected baby’s weight is between 3250 to 3251 grams. • To find probabilities of continuous random variables, we use probability distribution (or so called density) function.

7. Uniform Random Variable &Uniform Probability Distribution

8. Uniform Random Variable • Sometimes we want to model a continuous random variable that is equally likely between two limits • Examples • Choose a random time … the number of seconds past the minute is random number in the interval from 0 to 60 • Observe a tire rolling at a high rate of speed … choose a random time … the angle of the tire valve to the vertical is a random number in the interval from 0 to 360

9. Uniform Probability Distribution • When “every number” is equally likely in an interval, this is a uniformprobabilitydistribution • Any specific number has a zero probability of occurring • The mathematically correct way to phrase this is that any two intervals of equal length have the same probability

10. Example • For the seconds after the minute example • Every interval of length 3 has probability 3/60 • The chance that it will be between 14.4 and 17.3 seconds after the minute is 3/60 • The chance that it will be between 31.2 and 34.2 seconds after the minute is 3/60 • The chance that it will be between 47.9 and 50.9 seconds after the minute is 3/60

11. Probability Density Function • A probabilitydensityfunction is an equation used to specify and compute probabilities of a continuous random variable • This equation must have two properties • The total area under the graph of the equation is equal to 1 (the total probability is 1) • The equation is always greater than or equal to zero (probabilities are always greater than or equal to zero)

12. Probability Density Function • This function method is used to represent the probabilities for a continuous random variable • For the probability of X between two numbers • Compute the area under the curve between the two numbers • That is the probability

13. The probability To 8 (here) From 4 (here) Area is the Probability • The probability of being between 4 and 8

14. Probability Density Function • An interpretation of the probability density function is • The random variable is more likely to be in those regions where the function is larger • The random variable is less likely to be in those regions where the function is smaller • The random variable is never in those regions where the function is zero

15. Less likely values More likely values Probability Density Function • A graph showing where the random variable has more likely and less likely values

16. Uniform Probability Density Function • The time example … uniform between 0 and 60 • All values between 0 and 60 are equally likely, thus the equation must have the same value between 0 and 60

17. Uniform Probability Density Function • The time example … uniform between 0 and 60 • Values outside 0 and 60 are impossible, thus the equation must be zero outside 0 to 60

18. 1/60 Uniform Probability Density Function • The time example … uniform between 0 and 60 • Because the total area must be one, and the width of the rectangle is 60, the height must be 1/60. Therefore the uniform probability density is a constant ( the equation is )

19. 1/60 Uniform Probability Density Function • The time example … uniform between 0 and 60 • The probability that the variable is between two numbers is the area under the curve between them

20. Normal Random Variable &Normal Probability Distribution

21. Overview • The normal distribution models bell shaped variables • The normal distribution is the fundamental distribution underlying most of inferential statistics

22. Chapter 7 – Section 1 • The normalcurve has a very specific bell shaped distribution • The normal curve looks like

23. Normal Random Variable • A normallydistributed random variable, or a variable with a normalprobabilitydistribution, is a random variable that has a relative frequency histogram in the shape of a normal curve • This curve is also called the normaldensitycurve/function or normal curve (a particular probability density function) • The normal distribution models bell shaped variables • The normal distribution is the fundamental distribution underlying most of inferential statistics

24. Normal Density Curve • In drawing the normal curve, the mean μ and the standard deviation σ have specific roles • The mean μ is the center of the curve • The values (μ – σ) and (μ + σ) are the inflection points of the curve, where the concavity of the curve changes.

25. Normal Density Curve • There are normal curves for each combination of μ and σ • The curves look different, but the same too • Different values of μ shift the curve left and right • Different values of σ shift the curve up and down

26. Normal Curve • Two normal curves with different means (but the same standard deviation) • The curves are shifted left and right

27. Normal Density Curve • Two normal curves with different standard deviations (but the same mean) • The curves are shifted up and down

28. Properties of Normal Curve • Properties of the normal density curve • The curve is symmetric about the mean • The mean = median = mode, and this is the highest point of the curve • The curve has inflection points at (μ – σ) and (μ + σ) • The total area under the curve is equal to 1. The total area is equal to 1. (It is complicated to show this. But it is true.) • The area under the curve to the left of the mean is equal to the area under the curve to the right of the mean

29. Properties of Normal Curve • Properties of the normal density curve • As x increases, the curve getting close to zero (never goes to zero, though)… as x decreases, the curve getting close to zero (never goes to zero) • In addition, • The area within 1 standard deviation of the mean is approximately 0.68 • The area within 2 standard deviations of the mean is approximately 0.95 • The area within 3 standard deviations of the mean is approximately 0.997 (almost 100%) This is so called empirical rule. Therefore, a normal curve will be close to zero at about 3 standard deviation below and above the mean.

30. Empirical Rule • The empirical rule or 68-95-99.7 rule is true • Approximately 68% of the values lie between(μ – σ) and (μ + σ) • Approximately 95% of the values lie between(μ – 2σ) and (μ + 2σ) • Approximately 99.7% of the values lie between(μ – 3σ) and (μ + 3σ) • These are difficult calculations, but they are true

31. Empirical Rule ( 68-95-99.7 Rule) • An illustration of the Empirical Rule

32. Histogram & Density Curve • When we collect data, we can draw a histogram to summarize the results • However, using histograms has several drawbacks • Histograms are grouped, so • There are always grouping errors • It is difficult to make detailed calculations

33. Histogram & Density Curve • Instead of using a histogram, we can use a probability density function that is an approximation of the histogram • Probability density functions are not grouped, so • There are not grouping errors • They can be used to make detailed calculations

34. Normal Histogram • Frequently, histograms are bell shaped such as • We can approximate these with normal curves

35. Normal Curve Approximation • Lay over the top of the histogram with a curve such as • In this case, the normal curve is close to the histogram, so the approximation should be accurate

36. Normal Density Probability Function • The equation of the normal curve with mean μ and standard deviation σ is • This is a complicated formula, but we will never need to use it for the calculation of probabilities. (thankfully)

37. Modeling with Normal Curve • When we model a distribution with a normal probability distribution, we use the area under the normal curve to • Approximate the areas of the histogram being modeled • Approximate probabilities that are too detailed to be computed from just the histogram

38. Example • Assume that the distribution of giraffe weights has μ = 2200 pounds and σ = 200 pounds

39. Example Continued • What is an interpretation of the area under the curve to the left of 2100?

40. Example Continued • It is the proportion of giraffes that weigh 2100 pounds and less Note: Area = Probability = Proportion

41. Standardize Normal Random Variable • How do we calculate the areas under a normal curve? • If we need a table for every combination of μ and σ, this would rapidly become unmanageable • We would like to be able to compute these probabilities using just one table • The solution is to use the standard normal random variable

42. Standard Normal Random Variable • The standard normal random variable is the specific normal random variable that has μ = 0 and σ = 1 • We can relate general normal random variables to the standard normal random variable using a so-called Z-score calculation

43. Standard Normal Random Variable • If X is a general normal random variable with mean μ and standard deviation σ then is a standard normal random variable ( Z-score) • This equation connects general normal random variables with the standard normal random variable • We only need a standard normal table

44. Example • The area to the left of 2100 for a normal curve with mean 2200 and standard deviation 200

45. Example Continued • To compute the corresponding value of Z, we use the Z-score • Thus the value of X = 2100 corresponds to a value of Z = – 0.5

46. Symmary • Normal probability distributions can be used to model data that have bell shaped distributions • Normal probability distributions are specified by their means and standard deviations • Areas under the curve of general normal probability distributions can be related to areas under the curve of the standard normal probability distribution

47. The Standard Normal Distribution

48. Objectives • Find the area under the standard normal curve • Find Z-scores for a given area • Interpret the area under the standard normal curve as a probability

49. How to Compute Area under Standard Normal Curve • There are several ways to calculate the area under the standard normal curve • We can use a table (such as Table IV on the inside back cover) • We can use technology (a calculator or software) • Using technology is preferred

50. Compute Area under Standard Normal Curve • Three different area calculations • Find the area to the left of • Find the area to the right of • Find the area between • Two different methods shown here • From a table • Using TI Graphing Calculator (recommended method)