1. Part A: Concepts & binomial distributions Part B: Normal distributions
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4: Probability
Part A: Concepts & binomial distributions
Part B: Normal distributions
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Unit 4: Intro to probability
Biostat

2. Unit 4: Intro to probability
Definitions
Random variable  a numerical quantity that takes on different values depending on chance
Population  the set of all possible values for a random variable
Event  an outcome or set of outcomes for a random variable
Probability  the proportion of times an event occurs in the population; (long-run) expected proportion
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Unit 4: Intro to probability

3. Probability (definition #1)
The probability of an event is its relative frequency (proportion) in the population.
Example:
Let A  selecting a female at random from an HIV+ population
There are 600 people in the population.
There are 159 females.
Therefore, Pr(A) = 159 ÷ 600 = 0.265
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Unit 4: Intro to probability

4. Probability (definition #2)
The probability of an event is its expected proportion when the process in repeated again and again under the same conditions
Select 100 individuals at random
24 are female
Pr(A)  24 ÷ 100 = 0.24
This is only an estimate (unless n is very very big)
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Unit 4: Intro to probability

5. Probability (definition #3)
The probability of an event is a quantifiable level of belief between 0 and 1
Probability
Verbal expression
0.00
Never
0.05
Seldom
0.20
Infrequent
0.50
As often as not
0.80
Very frequent
0.95
Highly likely
1.00
Always
Example:
Prior experience suggests a quarter of population is female. Therefore, Pr(A) ≈ 0.25
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Unit 4: Intro to probability

6. Some rules of probability
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Unit 4: Intro to probability

7. Types of random variables
Discrete have a finite set of possible outcomes,
e.g. number of females in a sample of size n (0, 1, 2, …, n)
We cover binomial random variables
Continuous have a continuum of possible outcomes
e.g., average body weight (lbs) in a sample (160, 160.5, , , …)
We cover Normal random variables
There are other random variable families, but only binomial and Normal RVs are covered for now.
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Unit 4: Intro to probability

8. Binomial distributions
Most popular type of discrete RV
Based on Bernoulli trial  random event characterized by “success” or “failure”
Examples
Coin flip (heads or tails)
Survival (yes or no)
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Unit 4: Intro to probability

9. Binomial random variables
Binomial random variable  random number of successes in n independent Bernoulli trials
A family of distributions identified by two parameters
n  number of trials
p  probability of success for each trial
Notation: X~b(n,p)
X  random variable
~  “distributed as”
b(n, p)  binomial RV with parameters n and p
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Unit 4: Intro to probability

10. “Four patients” example
A treatment is successful 75% of time
We treat 4 patients
X  random number of successes, which varies  0, 1, 2, 3, or 4 depending on binomial distribution X~b(4, 0.75)
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Unit 4: Intro to probability

11. The probability of i successes is …
Binomial formula
The probability of i successes is …
Where
nCi = the binomial coefficient (next slide)
p = probability of success for each trial
q = probability of failure = 1 – p
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Unit 4: Intro to probability

12. Binomial coefficient (“choose function”)
where
!  the factorial function: x! = x  (x – 1)  (x – 2)  …  1
Example: 4! = 4  3  2  1 = 24 By definition 1! = 1 and 0! = 1
nCi  the number of ways to choose i items out of n
Example: “4 choose 2”:
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Unit 4: Intro to probability

13. “Four patients” example
n = 4 and p = 0.75 (so q = = 0.25)
Question: What is probability of 0 successes?  i = 0
Pr(X = 0) =nCi pi qn–i = 4C0 · · 0.254–0 = 1 · · =
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Unit 4: Intro to probability

14. Unit 4: Intro to probability
X~b(4,0.75), continued
Pr(X = 1) = 4C1 · · –1
= 4 · · =
Pr(X = 2) = 4C2 · · –2
= 6 · · =
(Do not demonstrate all calculations. Students should prove to themselves they derive and interpret these values.)
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Unit 4: Intro to probability

15. Unit 4: Intro to probability
X~b(4, 0.75) continued
Pr(X = 3) = 4C3 · · –3
= 4 · · 0.25 =
Pr(X = 4) = 4C4 · · –4
= 1 · · 1 =
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Unit 4: Intro to probability

16. The distribution X~b(4, 0.75)
Probability table for X~b(4,.75)
Probability curve for X~b(4,.75)
Successes
Probability
0.0039 1
0.0469 2
0.2109 3
0.4210 4
0.3164
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Unit 4: Intro to probability

17. Area under the curve (AUC) concept
The area under a probability curve (AUC) = probability!
Get it?
Pr(X = 2) = .2109
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Unit 4: Intro to probability

18. Cumulative probability (left tail)
Cumulative probability = Pr(X  i) = probability less than or equal to i
Illustrative example: X~b(4, .75)
Pr(X  0) = Pr(X = 0) = .0039
Pr(X  1) = Pr(X  0) + Pr(X = 1) = =
Pr(X  2) = Pr(X  1) + Pr(X = 2) = =
Pr(X  3) = Pr(X  2) + Pr(X = 3) = =
Pr(X  4) = Pr(X  3) + Pr(X = 4) = =
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Unit 4: Intro to probability

19. Unit 4: Intro to probability
X~b(4, 0.75)
Probability function
Cumulative probability
Pr(X  0)
0.0039
Pr(X  1)
0.0469
0.0508
Pr(X  2)
0.2109
0.2617
Pr(X  3)
0.4210
0.6836
Pr(X  4)
0.3164
1.0000
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Unit 4: Intro to probability

20. Cumulative probability
left tail = cumulative probability
Area under shaded bars in left tail sums to , i.e., Pr(X  2) =
Area under “curve” = probability
Bring it on!
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Unit 4: Intro to probability

21. Reasoning
Use probability model to reasoning about chance.
I hypothesize p = 0.75, but observe only 2 successes. Should I doubt my hypothesis?
ANS: No. When p = 0.75, you’ll see 2 or fewer successes 25% of the time (not that unusual).
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Unit 4: Intro to probability

22. StaTable probability calculator
Link on course homepage
Three versions
Java (browser)
Windows
Palm
Probability
Cumulative probability
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Unit 4: Intro to probability

23. Intro to Probability, Part B
The Normal distributions
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Unit 4: Intro to probability

24. The Normal distributions
Most popular continuous model
Recognized by de Moivre (1667– 1754)
Extended by Laplace (1749 – 1827)
How’s my hair?
Looks good.
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Unit 4: Intro to probability

25. Probability density function (curve)
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Probability density function (curve)
Example: vocabulary scores of 947 seventh graders
Smooth curve drawn over histogram is a model of the actual distribution
Mathematical model is the Normal probability density function (pdf)
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Unit 4: Intro to probability
Biostat

26. Unit 4: Intro to probability
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Area under curve
The area under the curve (AUC) concepts applies
The shaded bars (left tail) represent scores ≤ 6.0 = 30.3% of scores
Pr(X ≤ 6) = 0.303
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Unit 4: Intro to probability
Biostat

27. Areas under curve (cont.)
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Areas under curve (cont.)
Now translate this to the area under the curve (AUC)
The scale of the Y-axis is adjusted so the total AUC = 1
The AUC to the left of 6.0 (shaded) = 0.293
Therefore, the AUC “models” the area in proportion area in the bars of the histogram, i.e., probabilities of associated ranges
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Unit 4: Intro to probability
Biostat

28. Unit 4: Intro to probability
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Density Curves
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Unit 4: Intro to probability
Biostat

29. Arrows indicate points of inflection
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Normal distributions
Normal distributions = a family of distributions with common characteristics
Normal distributions have two parameters
Mean µ locates center of the curve
Standard deviation  quantifies spread (at points of inflection)
Arrows indicate points of inflection
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Unit 4: Intro to probability
Biostat

30. Unit 4: Intro to probability
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rule for Normal RVs
68% of AUC falls within 1 standard deviation of the mean (µ  )
95% fall within 2 (µ  2)
99.7% fall within 3 (µ  3)
11/13/2018
Unit 4: Intro to probability
Biostat

31. Illustrative example: WAIS
Wechsler adult intelligence scores (WAIS) vary according to a Normal distribution with μ = 100 and σ = 15
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Unit 4: Intro to probability

32. Another example (male height)
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Another example (male height)
Adult male height is approximately Normal with µ = 70.0 inches and  = 2.8 inches (NHANES, 1980)
Shorthand: X ~ N(70, 2.8)
Therefore:
68% of heights = µ   = 70.0  2.8 = 67.2 to 72.8
95% of heights = µ  2 = 70.0  2(2.8) = 64.4 to 75.6
99.7% of heights = µ  3 = 70.0  3(2.8) = 61.6 to 78.4
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Unit 4: Intro to probability
Biostat

33. Another example (male height)
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Another example (male height)
What proportion of men are less than 72.8 inches tall? (Note: 72.8 is one σ above μ) ?
(height)
68%
(by Rule) -1 +1
16%
16%
84%
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Unit 4: Intro to probability
Biostat

34. Male Height Example ? 68 70 (height)
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Male Height Example
What proportion of men are less than 68 inches tall? ?
(height)
68 does not fall on a ±σ marker.
To determine the AUC, we must first standardize the value.
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Unit 4: Intro to probability
Biostat

35. Standardized value = z score
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Standardized value = z score
To standardize a value, simply subtract μ and divide by σ
This is now a z-score
The z-score tells you the number of standard deviations the value falls from μ
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Unit 4: Intro to probability
Biostat

36. Example: Standardize a male height of 68”
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Example: Standardize a male height of 68”
Recall X ~ N(70,2.8)
Therefore, the value 68 is 0.71 standard deviations below the mean of the distribution
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Unit 4: Intro to probability
Biostat

37. Men’s Height (NHANES, 1980) ? 68 70 (height values)
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Men’s Height (NHANES, 1980)
What proportion of men are less than 68 inches tall? =
What proportion of a Standard z curve is less than –0.71?
(height values) ?
(standardized values)
You can now look up the AUC in a
Standard Normal “Z” table.
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Unit 4: Intro to probability
Biostat

38. Using the Standard Normal table
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Using the Standard Normal table z
.00
.01
.02
0.8
.2119
.2090
.2061
0.7
.2420
.2389
.2358
0.6
.2743
.2709
.2676
Pr(Z ≤ −0.71) = .2389
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Unit 4: Intro to probability
Biostat

39. Summary (finding Normal probabilities)
Draw curve w/ landmarks
Shade area
Standardize value(s)
Use Z table to find appropriate AUC
(standardized values)
(height values)
.2389
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Unit 4: Intro to probability

40. Right-”tail” 68 70 (height values)
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Right-”tail”
What proportion of men are greater than 68” tall?
Greater than  look at right “tail”
Area in right tail = 1 – (area in left tail)
(standardized values)
(height values)
.2389
= .7611
Therefore, 76.11% of men are greater than 68 inches tall.
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Unit 4: Intro to probability
Biostat

41. Unit 4: Intro to probability
Z percentiles
zp  the z score with cumulative probability p
What is the 50th percentile on Z? ANS: z.5 = 0
What is the 2.5th percentile on Z? ANS: z.025 = 2
What is the 97.5th percentile on Z? ANS: z.975 = 2
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Unit 4: Intro to probability

42. Finding Z percentile in the table
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Finding Z percentile in the table
Look up the closest entry in the table
Find corresponding z score
e.g., What is the 1st percentile on Z?
z.01 = -2.33
closest cumulative proportion is .0099 z
.02
.03
.04
2.3
.0102
.0099
.0096
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Unit 4: Intro to probability
Biostat

43. Unstandardizing a value
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Unstandardizing a value
How tall must a man be to place in the lower 10% for men aged 18 to 24?
.10
? (height values)
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Unit 4: Intro to probability
Biostat

44. Table A: Standard Normal Table
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Table A: Standard Normal Table
Use Table A
Look up the closest proportion in the table
Find corresponding standardized score
Solve for X (“un-standardize score”)
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Unit 4: Intro to probability
Biostat

45. Table A: Standard Normal Proportion
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Table A: Standard Normal Proportion z
.07
.09
1.3
.0853
.0838
.0823
.1020
.0985
1.1
.1210
.1190
.1170
.08
1.2
.1003
Pr(Z < -1.28) = .1003
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Unit 4: Intro to probability
Biostat

46. Men’s Height Example (NHANES, 1980)
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Men’s Height Example (NHANES, 1980)
How tall must a man be to place in the lower 10% for men aged 18 to 24?
.10
? (height values)
(standardized values)
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Unit 4: Intro to probability
Biostat

47. Observed Value for a Standardized Score
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Observed Value for a Standardized Score
“Unstandardize” z-score to find associated x :
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Unit 4: Intro to probability
Biostat

48. Observed Value for a Standardized Score
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Observed Value for a Standardized Score
x = μ + zσ = 70 + (-1.28 )(2.8) = 70 + (3.58) = 66.42
A man would have to be approximately inches tall or less to place in the lower 10% of the population
11/13/2018
Unit 4: Intro to probability
Biostat