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Chapter 8 Random Variables

Chapter 8 Random Variables. Probability Distributions Discrete Random Variables Continuous Random Variables Binomial Probability Model. Randomness. A random variable is a numerical measurement of the outcome of a random phenomenon. It is a quantity that can take on different values.

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Chapter 8 Random Variables

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  1. Chapter 8Random Variables • Probability Distributions • Discrete Random Variables • Continuous Random Variables • Binomial Probability Model

  2. Randomness • A random variable is a numerical measurement of the outcome of a random phenomenon. It is a quantity that can take on different values. • Often, the randomness results from • selecting a random sample for a population or • performing a randomized experiment

  3. Random Variable • Use letters near the end of the alphabet, such as x, to symbolize: • variables • a particular value of the random variable • Use a capital letter, such as X, to refer to the random variable itself. • Example: Flip a coin three times • X= number of heads in the 3 flips; defines the random variable • x=2; represents a possible value of the random variable

  4. Probability Distribution • The probability distributionof a random variable specifies its possible values and their probabilities. • Note: It is the randomness of the variable that allows us to specify probabilities for the outcomes.

  5. Probability Distribution of a Discrete Random Variable • A discrete random variableX takes a set of separate values (such as 0,1,2,…) as its possible outcomes. • Its probability distribution assigns a probability P(x)to each possible value x: • For each x, the probability P(x)falls between 0 and 1. • The sum of the probabilities for all the possible x • values equals 1.

  6. Probability Distribution of a Discrete Random Variable Let X = the discrete random variable. k = a number the discrete r.v. could assume. P(X =k) is the probability that X equals k. • Probability distribution function (pdf) ofX is a table or rule that assigns probabilities to possible values of X.

  7. Example: Number of Courses • 35% of students taking four courses, • 45% taking five, • and remaining 20% are taking six courses. • Let X = number of courses a randomly selected student is taking • The probability distribution function of X can be given by:

  8. Example: Number of Courses

  9. Example: Number of Girls • Family has 3 children. Probability of a girl is ½.What are the probabilities of having 0, 1, 2, or 3 girls? Sample Space: For each birth, write either B or G. There are eight possible arrangements of B and G for three births. These are the simple events. Sample Space and Probabilities: The eight simple events are equally likely. Random Variable X: number of girls in three births. For each simple event, the value of X is the number of G’s listed.

  10. Example: Number of Girls • Probability Distribution Table

  11. Example: Number of Girls • Probability DistributionFunction for Number of Girls X: • Graph of the pdf of X:

  12. Example: Number of Home Runs in a Game • What is the estimated probability of at least three home runs? Table 8.1 Probability Distribution of Number of Home Runs in a Game for San Francisco Giants

  13. Example: Number of Home Runs in a Game The probability of at least three home runs in a game is P(3)+P(4)+P(5 or more)= 0.0556 + 0.0185 + 0 = 0.0741 Table 8.1 Probability Distribution of Number of Home Runs in a Game for San Francisco Giants

  14. Cumulative Distribution Function for a Discrete Random Variable Cumulative distribution function (cdf) for a random variable Xis a rule or table that provides the probabilities P(X ≤ k) for any real number k. Cumulative probability = probability that Xis less than or equal to a particular value.

  15. Cumulative Distribution Function for Number of Girls

  16. Example: A Mixture of Children • What is the probability that a family with 3 children will have at least one child of each sex? • If X = Number of Girls then either family has one girl and two boys (X = 1) or two girls and one boy (X = 2). • P(X = 1 or X = 2) = P(X = 1) + P(X = 2) • = 3/8 + 3/8 = 6/8 = 3/4 • Probability distribution for number of girls:

  17. Mean of a Discrete Probability Distribution • The mean of a probability distribution for a discrete random variable is: • where the sum is taken over all possible values of x. • The mean of a probability distribution is denoted by the parameter, . • The mean is a weighted average; values of x that are more likely receive greater weight P(x).

  18. Expected Value of X • The mean of a probability distribution of a random variable X is also called the expected value of X. • The expected value reflects not what we’ll observe in a single observation, but rather what we expect for the average in a long run of observations. • It is not unusual for the expected value of a random variable to equal a number that is NOT a possible outcome.

  19. Example: Number of Home Runs in a Game • Find the mean of this probability distribution. Table 8.1 Probability Distribution of Number of Home Runs in a Game for San Francisco Giants

  20. Example: Number of Home Runs in a Game • The mean: • = 0(0.3889) + 1(0.3148) + 2(0.2222) + 3(0.0556) + 4(0.0185) • = 0 * P(0) + 1 * P(1) + 2 * P(2) + 3 * P(3) + 4 * P(4) • = 1

  21. The Standard Deviation of a Discrete Random Variable • The standard deviation of a probability distribution, denoted by the parameter, , measures variability from the mean. • Larger values of correspond to greater spread. • Roughly, describes how far the random variable falls, on the average, from the mean of its distribution.

  22. Standard Deviation of a Discrete Random Variable • The standard deviation of a random variable is roughly the average distance the random variable falls from its mean, or expected value, over the long run. • If X is a random variable with possible values x1, x2, x3, . . . , occurring with probabilities p1, p2, p3, . . . , and expected value E(X) = m, then

  23. Example: Stability or Excitement? • Two plans for investing $100 – which would you choose?

  24. Example: Stability or Excitement? Expected Value For Each Plan: Plan 1:E(X ) = $5,000(.001) + $1,000(.005) + $0(.994) = $10.00 Plan 2: E(Y ) = $20(.3) + $10(.2) + $4(.5) = $10.00

  25. Example: Stability or Excitement? • Variability for Each Plan:

  26. Example: Stability or Excitement? Plan 1: V(X ) = $29,900.00 and s = $172.92 Plan 2: V(X ) = $48.00 and s = $6.93 • The possible outcomes for Plan 1 are much more variable. If you wanted to invest cautiously, you would choose Plan 2, but if you wanted to have the chance to gain a large amount of money, you would choose Plan 1.

  27. More Examples From Midterm II Practice Sheet

  28. Continuous Random Variable • A continuous random variable has an infinite continuum of possible values in an interval. • Examples are: time, age and size measures such as height and weight. • Continuous variables are usually measured in a discrete manner because of rounding.

  29. Probability Distribution of a Continuous Random Variable • A continuous random variable has possible values that form an interval. Its probability distribution is specified by a curve. • Each interval has probability between 0 and 1. • The interval containing all possible values has probability equal to 1.

  30. Probability Distribution of a Continuous Random Variable Smooth curve approximation Figure 8.2 Probability Distribution of Commuting Time. The area under the curve for values higher than 45 is 0.15. Question: Identify the area under the curve represented by the probability that commuting time is less than 15 minutes, which equals 0.29.

  31. Useful Probability Relationships

  32. Useful Probability Relationships

  33. Useful Probability Relationships

  34. Useful Probability Relationships

  35. Bell-Shaped Distributions • Probabilities for Bell – Shaped Distributions or Continuous Random Variables

  36. Normal Distribution • The normal distribution is symmetric, bell-shaped and characterized by its mean and standard deviation . • The normal distribution is the most important distribution in statistics. • Many distributions have an approximately normal distribution. • The normal distribution also can approximate many discrete distributions well when there are a large number of possible outcomes. • Many statistical methods use it even when the data are not bell shaped.

  37. Normal Distribution • Normal distributions are • Bell shaped • Symmetric around the mean • The mean ( ) and the standard deviation ( ) completely describe the density curve. • Increasing/decreasing moves the curve along the horizontal axis. • Increasing/decreasing controls the spread of the curve.

  38. Normal Distribution • Within what interval do almost all of the men’s heights fall? Women’s height? Figure 8.4 Normal Distributions for Women’s Height and Men’s Height. For each different combination of and values, there is a normal distribution with mean and standard deviation . Question: Given that = 70 and = 4, within what interval do almost all of the men’s heights fall?

  39. Empirical Rule or 68-95-99.7 Rule for Any Normal Curve • ≈ 68%of the observations fall within one standard deviation of the mean. • ≈ 95%of the observations fall within two standard deviations of the mean. • ≈ 99.7% of the observations fall within three standard deviations of the mean. Figure 8.5 The Normal Distribution. The probability equals approximately 0.68 within 1 standard deviation of the mean, approximately 0.95 within 2 standard deviations, and approximately 0.997 within 3 standard deviations. Question: How do these probabilities relate to the empirical rule?

  40. Empirical Rule or 68 – 95 – 99.7% Rule

  41. Example : 68-95-99.7% Rule • Heights of adult women can be approximated by a normal • distribution, inches; inches • 68-95-99.7 Rule for women’s heights: • 68% are between 61.5 and 68.5 inches • 95% are between 58 and 72 inches 99.7% are between 54.5 and 75.5 inches

  42. Z-Scores and the Standard Normal Distribution • The z-score for a value x of a random variable is the number of standard deviations that x falls from the mean. • A negative (positive) z-score indicates that the value is below (above) the mean. • Z-scores can be used to calculate the probabilities of a normal random variable using the normal tables in the back of the book.

  43. Standard Score (z – score) • The formula for converting any value x to a z-score is • . • A z-score measures the number of standard deviations that a value falls from the mean

  44. Shifting and Rescaling Data • Shifting data: • Adding (or subtracting) a constant to every data value adds (or subtracts) the same constant to measures of position. • Adding (or subtracting) a constant to each value will increase (or decrease) measures of position: center, percentiles, max or min by the same constant. • Its shape and spread - range, IQR, standard deviation - remain unchanged.

  45. Shifting and Rescaling Data • The following histograms show a shift from men’s actual weights to kilograms above recommended weight:

  46. Shifting and Rescaling Data • Rescaling data: • When we multiply (or divide) all the data values by any constant, all measures of position (such as the mean, median, and percentiles) and measures of spread (such as the range, the IQR, and the standard deviation) are multiplied (or divided) by that same constant.

  47. Shifting and Rescaling Data • The men’s weight data set measured weights in kilograms. If we want to think about these weights in pounds, we would rescale the data:

  48. Examples: Midterm Exam II Practice Sheet

  49. Z-Scores and the Standard Normal Distribution • A standard normal distribution has mean and standard deviation . • When a random variable has a normal distribution and its values are converted to z-scores by subtracting the mean and dividing by the standard deviation, the z-scores follow the standard normal distribution.

  50. Table Z: Standard Normal Probabilities • Table Z enables us to find normal probabilities. • It tabulates the normal cumulative probabilities falling below the point . • To use the table: • Find the corresponding z-score. • Look up the closest standardized score (z) in the table. • First column gives z to the first decimal place. • First row gives the second decimal place of z. • The corresponding probability found in the body of the table gives the probability of falling below the z-score.

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