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Random Variables

Learn about random variables, discrete and continuous variables, probability distributions, mean and variance, and the binomial distribution.

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Random Variables

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  1. Random Variables Numerical Quantities whose values are determine by the outcome of a random experiment

  2. Discrete Random Variables Discrete Random Variable: A random variable usually assuming an integer value. • a discrete random variable assumes values that are isolated points along the real line. That is neighbouring values are not “possible values” for a discrete random variable Note: Usually associated with counting • The number of times a head occurs in 10 tosses of a coin • The number of auto accidents occurring on a weekend • The size of a family

  3. Continuous Random Variables Continuous Random Variable: A quantitative random variable that can vary over a continuum • A continuous random variable can assume any value along a line interval, including every possible value between any two points on the line Note: Usually associated with a measurement • Blood Pressure • Weight gain • Height

  4. Probability Distributionsof a Discrete Random Variable

  5. Probability Distribution & Function Probability Distribution:A mathematical description of how probabilities are distributed with each of the possible values of a random variable. Notes: • The probability distribution allows one to determine probabilities of events related to the values of a random variable. • The probability distribution may be presented in the form of a table, chart, formula. Probability Function: A rule that assigns probabilities to the values of the random variable

  6. Comments: 1. The probability assigned to each value of the random variable must be between 0 and 1, inclusive: 2. The sum of the probabilities assigned to all the values of the random variable must equal 1: 3. Every probability function must satisfy:

  7. Mean and Variance of aDiscrete Probability Distribution • Describe the center and spread of a probability distribution • The mean (denoted by greek letter m (mu)), measures the centre of the distribution. • The variance (s2) and the standard deviation (s) measure the spread of the distribution. • s is the greek letter for s.

  8. Mean of a Discrete Random Variable • The mean, m, of a discrete random variable x is found by multiplying each possible value of x by its own probability and then adding all the products together: Notes: • The mean is a weighted average of the values of X. • The mean is the long-run average value of the random variable. • The mean is centre of gravity of the probability distribution of the random variable

  9. Variance and Standard Deviation 2 s = s Variance of a Discrete Random Variable: Variance, s2, of a discrete random variable x is found by multiplying each possible value of the squared deviation from the mean, (x-m)2, by its own probability and then adding all the products together: Standard Deviation of a Discrete Random Variable: The positive square root of the variance:

  10. Example The number of individuals, X, on base when a home run is hit ranges in value from 0 to 3.

  11. Computing the mean: Note: • 0.929 is the long-run average value of the random variable • 0.929 is the centre of gravity value of the probability distribution of the random variable

  12. Computing the variance: • Computing the standard deviation:

  13. The Binomial distribution • We have an experiment with two outcomes – Success(S) and Failure(F). • Let p denote the probability of S (Success). • In this case q=1-p denotes the probability of Failure(F). • This experiment is repeated n times independently. • X denote the number of successes occuring in the n repititions.

  14. The possible values of X are 0, 1, 2, 3, 4, … , (n – 2), (n – 1), n and p(x) for any of the above values of x is given by: X is said to have the Binomial distribution with parameters n and p.

  15. Summary: X is said to have the Binomial distribution with parameters n and p. • X is the number of successes occurring in the n repetitions of a Success-Failure Experiment. • The probability of success is p. • The probability function

  16. Example: • A coin is tossed n = 5 times. X is the number of heads occurring in the 5tosses of the coin. In this case p = ½ and

  17. Note:

  18. Computing the summary parameters for the distribution – m, s2, s

  19. Computing the mean: • Computing the variance: • Computing the standard deviation:

  20. Example: • A surgeon performs a difficult operation n = 10 times. • X is the number of times that the operation is a success. • The success rate for the operation is 80%. In this case p = 0.80 and • X has a Binomial distribution with n = 10 and p = 0.80.

  21. Computing p(x) for x = 1, 2, 3, … , 10

  22. The Graph

  23. Computing the summary parameters for the distribution – m, s2, s

  24. Computing the mean: • Computing the variance: • Computing the standard deviation:

  25. Notes • The value of many binomial probabilities are found in Tables posted on the Stats 244 site. • The value that is tabulated for n = 1, 2, 3, …,20; 25 and various values of p is: • Hence • The other table, tabulates p(x). Thus when using this table you will have to sum up the values

  26. Example • Suppose n = 8 and p = 0.70 and we want to compute P[X = 5] = p(5) • Table value for n = 8, p = 0.70 and c =5 is 0.448 = P[X 5] • P[X = 5] = p(5) = P[X 5] - P[X 4] = 0.448 – 0.194 = .254

  27. We can also compute Binomial probabilities using Excel The function =BINOMDIST(x, n, p, FALSE) will compute p(x). The function =BINOMDIST(c, n, p, TRUE) will compute

  28. Mean,Variance & Standard Deviation • The mean, variance and standard deviation of the binomial distribution can be found by using the following three formulas:

  29. Example Solutions: 1) n = 20, p = 0.75, q = 1 - 0.75 = 0.25 m = = = np ( 20 )(0. 75 ) 15 s = = = » npq ( 20 )(0. 75 )(0. 25 ) 3 . 75 1 . 936 2) These values can also be calculated using the probability function: 20 æ ö - x 20 x = = p ( x ) (0. 75 ) (0. 25 ) for x 0, 1, 2, . . . , 20 ç ÷ è ø x • Example: Find the mean and standard deviation of the binomial distribution when n = 20 and p = 0.75

  30. Table of probabilities

  31. Computing the mean: • Computing the variance: • Computing the standard deviation:

  32. Histogram m s

  33. Probability Distributionsof Continuous Random Variables

  34. Probability Density Function The probability distribution of a continuousrandom variable is describe by probability density curve f(x).

  35. Notes: • The Total Area under the probability density curve is 1. • The Area under the probability density curve is from a to b is P[a < X < b].

  36. Normal Probability Distributions

  37. Normal Probability Distributions • The normal probability distribution is the most important distribution in all of statistics • Many continuous random variables have normal or approximately normal distributions

  38. The Normal Probability Distribution Points of Inflection

  39. Main characteristics of the Normal Distribution • Bell Shaped, symmetric • Points of inflection on the bell shaped curve are at m – s and m + s. That is one standard deviation from the mean • Area under the bell shaped curve between m – s and m + s is approximately 2/3. • Area under the bell shaped curve between m – 2s and m + 2s is approximately 95%.

  40. Normal m = 100, s =20 Normal m = 100, s = 40 Normal m = 140, s =20 There are many Normal distributions depending on by m and s

  41. The Standard Normal Distributionm = 0, s = 1

  42. There are infinitely many normal probability distributions (differing in m and s) • Area under the Normal distribution with mean m and standard deviation s can be converted to area under the standard normal distribution • If X has a Normal distribution with mean m and standard deviation s than has a standard normal distribution has a standard normal distribution. • z is called the standard score (z-score) of X.

  43. Example: Suppose a man aged 40-45 is selected at random from a population. • X is the Blood Pressure of the man. • X is random variable. • Assume that X has a Normal distribution with mean m =180 and a standard deviation s = 15.

  44. The probability density of X is plotted in the graph below. • Suppose that we are interested in the probability that X between 170 and 210.

  45. Let Hence

  46. Standard Normal Distribution Properties: • The total area under the normal curve is equal to 1 • The distribution is bell-shaped and symmetric; it extends indefinitely in both directions, approaching but never touching the horizontal axis • The distribution has a mean of 0 and a standard deviation of 1 • The mean divides the area in half, 0.50 on each side • Nearly all the area is between z = -3.00 and z = 3.00 Notes: • Normal Table, Posted on Stats 244 web site, lists the probabilities below a specific value of z • Probabilities of other intervals are found using the table entries, addition, subtraction, and the properties above

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