1. Random Variables Chapter 8
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2. 8.1 What is a Random Variable?
Random Variable: assigns a number to each outcome of a random circumstance, or, equivalently, to each unit in a population.
Two different broad classes of random variables:
A continuous random variable can take any value in an interval or collection of intervals.
A discrete random variable can take one of a countable list of distinct values.
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3. Example 8.1 Random Variables at an Outdoor Graduation or Wedding
Random factors that will determine how enjoyable the event is: Temperature: continuous random variable Number of airplanes that fly overhead: discrete random variable
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4. Example 8.2 Probability an Event Occurs Three Times in Three Tries
What is the probability that three tosses of a fair coin will result in three heads?
Assuming boys and girls are equally likely, what is the probability that 3 births will result in 3 girls?
Assuming probability is 1/2 that a randomly selected individual will be taller than median height of a population, what is the probability that 3 randomly selected individuals will all be taller than the median?
Answer to all three questions = 1/8.
Discrete Random Variable X = number of times the “outcome of interest” occurs in three independent tries.
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5. 8.2 Discrete Random Variables
X = the random variable.
k = a number the discrete r.v. could assume.
P(X = k) is the probability that X equals k.
Discrete random variable: can only result in a countable set of possibilities – often a finite number of outcomes, but can be infinite.
Example 8.3 It’s Possible to Toss Forever
Repeatedly tossing a fair coin, and define: X = number of tosses until the first head occurs Any number of flips is a possible outcome.
P(X = k) = (1/2)k
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6. Probability Distribution of a Discrete R.V.
Using the sample space to find probabilities:
Step 1: List all simple events in sample space.
Step 2: Find probability for each simple event.
Step 3: List possible values for random variable X and identify the value for each simple event.
Step 4: Find all simple events for which X = k, for each possible value k.
Step 5: P(X = k) is the sum of the probabilities for all simple events for which X = k.
Probability distribution function (pdf) X is a table or rule that assigns probabilities to possible values of X.
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7. Example 8.4 How Many Girls are Likely?
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.
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8. Example 8.4 How Many Girls? (cont)
Value of X for each simple event:
Probability distribution function for Number of Girls X:
Graph of the pdf of X:
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9. Conditions for Probabilities for Discrete Random Variables
Condition 1 The sum of the probabilities over all possible values of a discrete random variable must equal 1.
Condition 2 The probability of any specific outcome for a discrete random variable must be between 0 and 1.
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10. Cumulative Distribution Function of a Discrete Random Variable
Cumulative distribution function (cdf) for a random variable X is a rule or table that provides the probabilities P(X ≤ k) for any real number k.
Cumulative probability = probability that X is less than or equal to a particular value.
Example 8.4 Cumulative Distribution Function for the Number of Girls (cont)
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11. Finding Probabilities for Complex Events
Example 8.4 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
pdf for Number of Girls X:
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12. 8.3 Expectations for Random Variables
The expected value of a random variable is the mean value of the variable X in the sample space, or population, of possible outcomes.
If X is a random variable with possible values x1, x2, x3, , occurring with probabilities p1, p2, p3, , then the expected value of X is calculated as
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13. Example 8.6 California Decco Lottery
Player chooses one card from each of four suits. Winning card drawn from each suit. If one or more matches the winning cards => prize. It costs $1.00 for each play.
How much would you win/lose per ticket over long run?
=> Lose an average of 35 cents per play.
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14. Standard Deviation for a Discrete Random Variable
The standard deviation of a random variable is essentially the average distance the random variable falls from its mean 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
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15. Example 8.7 Stability or Excitement
Two plans for investing $ – which would you choose?
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
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16. Example 8.7 Stability or Excitement (cont)
Variability for each plan:
Plan 1: V(X ) = $29, 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.
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17. 8.3 Binomial Random Variables
Class of discrete random variables = Binomial -- results from a binomial experiment.
Conditions for a binomial experiment:
1. There are n “trials” where n is determined in advance and is not a random value.
2. Two possible outcomes on each trial, called “success” and “failure” and denoted S and F.
3. Outcomes are independent from one trial to the next.
4. Probability of a “success”, denoted by p, remains same from one trial to the next. Probability of “failure” is 1 – p.
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18. Examples of Binomial Random Variables
A binomial random variable is defined as X=number of successes in the n trials of a binomial experiment.
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19. Finding Binomial Probabilities
for k = 0, 1, 2, …, n
Example 8.9 Probability of Two Wins in Three Plays
p = probability win = 0.2; plays of game are independent.
X = number of wins in three plays.
What is P(X = 2)?
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20. Expected Value and Standard Deviation for a Binomial Random Variable
For a binomial random variable X based on n trials and success probability p,
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21. Example 8.12 Extraterrestrial Life?
50% of large population would say “yes” if asked, “Do you believe there is extraterrestrial life?”
Sample of n = 100 is taken.
X = number in the sample who say “yes” is approximately a binomial random variable.
In repeated samples of n=100, on average 50 people would say “yes”. The amount by which that number would differ from sample to sample is about 5.
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22. 8.5 Continuous Random Variables
Continuous random variable: the outcome can be any value in an interval or collection of intervals.
Probability density function for a continuous random variable X is a curve such that the area under the curve over an interval equals the probability that X is in that interval.
P(a  X  b) = area under density curve over the interval between the values a and b.
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23. Example 8.13 Time Spent Waiting for Bus
Bus arrives at stop every 10 minutes. Person arrives at stop at a random time, how long will s/he have to wait?
X = waiting time until next bus arrives.
X is a continuous random variable over 0 to 10 minutes.
Note: Height is so total area under the curve is (0.10)(10) = 1
This is an example of a Uniform random variable
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24. Example 8.13 Waiting for Bus (cont)
What is the probability the waiting time X was in the interval from 5 to 7 minutes?
Probability = area under curve between 5 and 7
= (base)(height) = (2)(.1) = .2
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25. 8.6 Normal Random Variables
If a population of measurements follows a normal curve, and if X is the measurement for a randomly selected individual from that population, then
X is said to be a normal random variable
X is also said to have a normal distribution
Any normal random variable can be completely characterized by its mean, m, and standard deviation, s.
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26. Example 8.14 College Women’s Heights
Data suggest the distribution of heights of college women modeled by a normal curve with mean m = 65 inches and standard deviation s = 2.7 inches.
Note: Tick marks given at the mean and at 1, 2, 3 standard deviations above and below the mean.
Empirical Rule values are exact characteristics of a normal curve model
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27. Standard Scores 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.
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28. Example Height (cont)
For a population of college women, the z-score corresponding to a height of 62 inches is
This z-score tells us that 62 inches is 1.11 standard deviations below the mean height for this population.
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29. Finding Probabilities for z-scores
Table A.1 = Standard Normal (z) Probabilities
Body of table contains P(Z  z*).
Left-most column of table shows algebraic sign, digit before the decimal place, the first decimal place for z*.
Second decimal place of z* is in column heading.
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30. More Finding Probabilities for z-scores
Table A.1 = Standard Normal (z) Probabilities
P(Z  -3.00) =.0013 (see in portion above)
P(Z  −2.59) = .0048
P(Z  1.31) = .9049
P(Z  2.00) = .9772
P(Z  -4.75) = (from in the extreme)
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31. Example 8.15 Probability Z > 1.31
P(Z > 1.31) = 1 – P(Z  1.31)
= 1 – = .0951
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32. Example 8.16 Probability Z is between –2.59 and 1.31
P(-2.59  Z  1.31)
= P(Z  1.31) – P(Z  -2.59)
= – = .9001
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33. Use z-scores to Solve General Problems
Example Height (cont)
What is the probability that a randomly selected college woman is 62 inches or shorter?
About 13% of college women are 62 inches or shorter.
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34. Use z-scores to Solve General Problems
Example Height (cont)
What proportion of college woman are taller than 68 inches?
About 13% of college women are taller than 68 inches.
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35. Finding Percentiles
If 25th percentile of pulse rates is 64 bpm, then 25% of pulse rates are below 64 and 75% are above 64. The percentile is 64 bpm, and the percentile ranking is 25%.
Step 1: Find z-score that has specified cumulative probability. Using Table A.1, find percentile rank in body of table and read off the z-score.
Step 2: Calculate the value of variable that has the z-score found in step 1: x = z*s + m.
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36. Example 8.17 75th Percentile of Systolic Blood Pressure
Blood Pressures are normal with mean 120 and standard deviation 10. What is the 75th percentile?
Step 1: Find closest z* with area of in body of Table A.1.
z = 0.67
Step 2: Calculate x = z*s + m
x = (0.67)(10) = or about 127.
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37. 8.7 Approximating Binomial Distribution Probabilities
If X is a binomial random variable based on n trials with success probability p, and n is large, then the random variable X is also approximately normal, with mean and standard deviation given as:
Conditions: Both np and n(1 – p) are at least 10.
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38. Example 8.18 Number of H in 30 Flips
X = number of heads in n = 30 flips of fair coin Binomial distribution with n = 30 and p = .5.
Distribution is bell-shaped and can be approximated by a normal curve.
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39. Example 8.19 Political Woes
Poll: n = 500 adults; 240 supported position
If 50% of all adults support position, what is the probability that a random sample of 500 would find 240 or fewer holding this position? P(X  240)
X is approximately normal with
Not unlikely to see 48% or less, even if 50% in population favor.
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40. 8.8 Sums, Differences, Combinations of R.V.s
A linear combination of random variables, X, Y, is a combination of the form:
L = aX + bY + …
where a, b, etc. are numbers – positive or negative.
Most common: Sum = X + Y Difference = X – Y
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41. Means of Linear Combinations
The mean of L is:
Mean(L) = a Mean(X) + b Mean(Y) + …
Most common:
Mean( X + Y) = Mean(X) + Mean(Y)
Mean(X – Y) = Mean(X) – Mean(Y)
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42. Variances of Linear Combinations
If X, Y, are independent random variables, then
Variance(L) = a2 Variance(X) + b2 Variance(Y) + …
Most common:
Variance( X + Y) = Variance(X) + Variance(Y)
Variance(X – Y) = Variance(X) + Variance(Y)
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43. Combining Independent Normal Random Variables
If X, Y, are independent normal random variables, then L = aX + bY + … is normally distributed.
In particular:
X + Y is normal with
X – Y is normal with
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44. Example 8.20 Will Meg Miss Her Flight?
X = Meg’s travel time is normal with mean 25 minutes and std dev 3 minutes.
Y = Airport time is normal with mean 15 minutes and std dev 2 minutes.
X and Y independent so T = X + Y = total time is normal with
If Meg leaves with 45 minutes before flight takes off, what is probability she will miss her flight?
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45. Combining Independent Binomial Random Variables
If X, Y, are independent binomial random variables with nX, nY, etc trials and all have same success probability p, then the sum X + Y + … is a binomial random variable with n = nX + nY + … and success probability p.
If the success probabilities differ, the sum is not a binomial random variable.
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46. Example 8.22 Donations Add Up
Fund drive: Three volunteers call potential donors. About 20% called make a donation, independent of who called. If volunteers make 10, 12 and 18 calls, respectively, what is the probability that they get at least ten donors?
X = # donors by volunteer 1; binomial n = 10, p = .2
Y = # donors by volunteer 2; binomial n = 12, p = .2
W = # donors by volunteer 3; binomial n = 18, p = .2
T = X + Y + W is binomial n = 40, p = .2
Using calculator or computer:
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