6 Chapter Continuous Probability Distributions
Continuous Probability Distribution
Probability Distribution for r.v. X is...?
Continuous random variable?
Continuous Probability Distribution (continued)
Requirement with probability dist?
If X is continuous, then...
Continuous Probability Distribution (continued)
Consider the interval (a, b) where:
- ab If a = b, then?
The interval is a single point and P(single point)=0.
We want prob. that X
(a, b)
P(a < X < b)... which is the area under the curve over the interval a to b.
Normal Distribution
aka Gaussian distribution Most useful and most commonly used Plays an important role in statistics b/c:
Numerous phenomena seem to follow the normal dist. or can be approx. by it. Can be used to approx. discrete prob. dist. Plays very important role in classical statistical inference
Normal Distribution (continued)
Properties
Normal curve is bell-shaped Mean Median Mode Perfectly symmetric about the mean (central value)
Normal Distribution (continued)
Characterized by:
, the population mean , the population standard deviation Normal probability density function:
1 x 2
2
f (x)
1 e 2
, for all x
Normal Distribution (continued)
Normal Distribution (continued)
Function allows us to determine the probability that x falls in a given interval through numerical integration
b b
f ( x )dx
a a
1 e 2
1 x 2
2
dx , for all x
Normal Distribution (continued)
Normal Distribution (continued)
Integration method difficult & time-consuming Pre-generated table would be appropriate Unfortunately, there are an infinite number of normal tables since there are an infinite number of combinations for and . Fortunately, ALL normal probability distributions can be represented in ONE table. Each value of X can be identified by the number of standard deviations it is away from the mean
Normal Distribution & Empirical Rule
Standard Normal Distribution
Standard normal distribution (Z) has:
mean ( ) equal to 0 standard deviation ( ) equal to 1 We write: Z ~ N(0, 1) which is read as "Z is normally distributed with = 0 and = 1.
Standard Normal Distribution (continued)
All normal probability distributions any X~N( , ) can be converted to a standard normal distribution Z~N(0,1) by standardization. If X~N( , ), X can be standardized to a z-score where Z~N(0,1) by
z
X
Standard Normal Distribution (continued)
Standard Normal Curve
Gives cumulative probabilities P( Z < z) To use the table:
Read the value of z down the leftmost column and across the top row Find the corresponding value of the probability in the body of the table
Cumulative Probabilities for the Standard Normal Distribution
Cumulative probability
z
0
z -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0
0.00 0.0013 0.0019 0.0026 0.0035 0.0047 0.0062 0.0082 0.0107 0.0139 0.0179 0.0228 0.0287 0.0359 0.0446 0.0548 0.0668 0.0808 0.0968 0.1151 0.1357 0.1587 0.1841 0.2119 0.2420 0.2743 0.3085 0.3446 0.3821 0.4207 0.4602 0.5000
0.01 0.0013 0.0018 0.0025 0.0034 0.0045 0.0060 0.0080 0.0104 0.0136 0.0174 0.0222 0.0281 0.0351 0.0436 0.0537 0.0655 0.0793 0.0951 0.1131 0.1335 0.1562 0.1814 0.2090 0.2389 0.2709 0.3050 0.3409 0.3783 0.4168 0.4562 0.4960
0.02 0.0013 0.0018 0.0024 0.0033 0.0044 0.0059 0.0078 0.0102 0.0132 0.0170 0.0217 0.0274 0.0344 0.0427 0.0526 0.0643 0.0778 0.0934 0.1112 0.1314 0.1539 0.1788 0.2061 0.2358 0.2676 0.3015 0.3372 0.3745 0.4129 0.4522 0.4920
0.03 0.0012 0.0017 0.0023 0.0032 0.0043 0.0057 0.0075 0.0099 0.0129 0.0166 0.0212 0.0268 0.0336 0.0418 0.0516 0.0630 0.0764 0.0918 0.1093 0.1292 0.1515 0.1762 0.2033 0.2327 0.2643 0.2981 0.3336 0.3707 0.4090 0.4483 0.4880
0.04 0.0012 0.0016 0.0023 0.0031 0.0041 0.0055 0.0073 0.0096 0.0125 0.0162 0.0207 0.0262 0.0329 0.0409 0.0505 0.0618 0.0749 0.0901 0.1075 0.1271 0.1492 0.1736 0.2005 0.2296 0.2611 0.2946 0.3300 0.3669 0.4052 0.4443 0.4840
0.05 0.0011 0.0016 0.0022 0.0030 0.0040 0.0054 0.0071 0.0094 0.0122 0.0158 0.0202 0.0256 0.0322 0.0401 0.0495 0.0606 0.0735 0.0885 0.1056 0.1251 0.1469 0.1711 0.1977 0.2266 0.2578 0.2912 0.3264 0.3632 0.4013 0.4404 0.4801
0.06 0.0011 0.0015 0.0021 0.0029 0.0039 0.0052 0.0069 0.0091 0.0119 0.0154 0.0197 0.0250 0.0314 0.0392 0.0485 0.0594 0.0721 0.0869 0.1038 0.1230 0.1446 0.1685 0.1949 0.2236 0.2546 0.2877 0.3228 0.3594 0.3974 0.4364 0.4761
0.07 0.0011 0.0015 0.0021 0.0028 0.0038 0.0051 0.0068 0.0089 0.0116 0.0150 0.0192 0.0244 0.0307 0.0384 0.0475 0.0582 0.0708 0.0853 0.1020 0.1210 0.1423 0.1660 0.1922 0.2206 0.2514 0.2843 0.3192 0.3557 0.3936 0.4325 0.4721
0.08 0.0010 0.0014 0.0020 0.0027 0.0037 0.0049 0.0066 0.0087 0.0113 0.0146 0.0188 0.0239 0.0301 0.0375 0.0465 0.0571 0.0694 0.0838 0.1003 0.1190 0.1401 0.1635 0.1894 0.2177 0.2483 0.2810 0.3156 0.3520 0.3897 0.4286 0.4681
0.09 0.0010 0.0014 0.0019 0.0026 0.0036 0.0048 0.0064 0.0084 0.0110 0.0143 0.0183 0.0233 0.0294 0.0367 0.0455 0.0559 0.0681 0.0823 0.0985 0.1170 0.1379 0.1611 0.1867 0.2148 0.2451 0.2776 0.3121 0.3483 0.3859 0.4247 0.4641
Cumulative Probabilities for the Standard Normal Distribution
z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
0.00 0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987
0.01 0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778 0.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982 0.9987
0.02 0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.9830 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982 0.9987
0.03 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988
0.04 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988
0.05 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989
0.06 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985 0.9989
0.07 0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989
0.08 0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9990
0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990
Standard Normal Table Examples
P(0 Z 1) P(Z 2.37) P(Z > -0.45) P(-1 Z 2.15) P(-1.6 Z 1.6) P(2 Z 3.5)
Standard Normal Table Examples
Standard Normal Table Examples
Standard Normal Table Examples
Normal Distribution Word Poblems
1. Make sure X~N
Identify and
2. Write out the prob. to be determined (pictures) 3. Standardize X 4. Use Std. Normal Table to find probability
Normal Distribution Word Problems Example 1
Individual weights of a variety of hand wrapped cigars are bell-shaped with a mean of 24 grams a and variance of 9 grams squared. If you randomly selected a cigar, what is the probability that the cigar weighs between 15.25 grams and 19.5 grams?
Normal Distribution Word Problems Example 1
Normal Distribution Word Problems Example 2
A toy manufacturer has found that the number of hours needed for an assembly line to produce 1000 toy products each day is approximately normal, with a mean of 4 hours and a variance of 4 hours squared. What is the probability that the 1000 toys will be produced in no more than 4.5 hours?
Normal Distribution Word Problems Example 2
Normal Distribution Word Problems Example 3
The grade point averages of a large population of college students are approximately normally distributed with a mean of 2.4 and a standard deviation of 0.8. What is the probability that a randomly selected student will have a grade point average in excess of 3.0?
Normal Distribution Word Problems Example 3
Normal Distribution Word Problems Example 4
A machine used to regulate the amount of dye dispensed for mixing shades of paint can be set so that it discharges an average of milliliters of dye per can of paint. The amount of dye discharged is known to have a normal distribution with variance equal to 0.160. If more than 6 milliliters of dye are discharged when making a particular shade of blue paint, the shade is unacceptable. Determine the setting of so that no more than 1% of the cans of paint will be unacceptable.
Normal Distribution Word Problems Example 4
Normal Distribution Fractiles
Given a probability and want to find the value(s) of the normal random variable that corresponds to that probability Example: Average daily production of a manufacturing plant is known to be normally distributed, and the manager wants to pay an incentive bonus to the crew when their level of production exceeds 85% of the daily production.
Normal Distribution Fractiles (continued)
We are not interested in the probability of an event We want to calculate the production level This type of problem is called a fractile. To solve, we just work backwards.
Normal Distribution Fractiles (continued)
With the example...
Suppose, you are given a normal pop. with =100 and =10. Find the value of X such that 85% of the X's are less than or equal to this value.
Normal Distribution Fractiles (continued)
Normal Distribution Fractiles (continued)
Locate prob. in middle of table Find corresponding z-score (use closest prob. in table)
Un-standardize
Normal Distribution Steps to Working Fractiles
1. 2. 3. 4. 5. 6. 7. 8. Identify a normal dist, and Identify the probability (picture) Determine the fractile Determine the prob. table value Look up prob. table value Find corresponding value of z Decide if z is + or Unstandardize z to get X
Normal Distribution Fractile Example 1
The length of time required to complete a college achievement test is found to be normally distributed with a mean of 70 minutes and a standard deviation of 12 minutes. When should the test be terminated if we wish to allow sufficient time for 90% of the students to complete the test?
Normal Distribution Fractile Example 1
Normal Distribution Fractile Example 2
A manufacturing plant utilizes 3000 electric light bulbs that have a length of life that is normally distributed with a mean of 500 hours and a standard deviation of 50 hours. To minimize the number of bulbs that burn out during operation hours, all the bulbs are replaced after a given period of operation. How often should the bulbs be replaced if we wish not more than 1% of the bulbs to burn out between replacement periods?
Normal Distribution Fractile Example 2
Normal Approximation to Binomial
Binomial Distibution
Sample of n discrete trials One of two possible outcomes (S/F) P(S) = p X, the random variable, # of "successes" in n trials P(X = x) using binomial formula
P( X
x)
n x p (1 p) n x
x
Normal Approximation to Binomial (continued)
If n is large, prob. are difficult to calculate Often times, we want P(X xi), P(Xxi), or P(xiXxj). Normal approximation to binomial is best for large sample (large n) ROT: Normal distribution should only be used to approximate the binomial distribution if n is greater than or equal to 5 divided by the minimum of p and (1 p).
Normal Approximation to Binomial (continued)
n 5 min( p, 1 p)
or (np, n (1 p)) 5
Normal Approximation to Binomial (continued)
Binomial Distribution
If n 5/min(p, q), then we can approximate the binomial distribution with a normal distribution with =np and 2=npq.
Normal Approximation to Binomial Example 1
Suppose we toss a coin 25 times and we want to find the probability that we get at least 10 but not more than 15 heads.
Normal Approximation to Binomial Example 1 (continued)
Is n large enough?
Normal Approximation to Binomial Example 1 (continued)
Continuity Correction
No area under the curve for X=10 or X=15 Assign interval under the curve from 9.5 to 15.5, so points 10, 11, 12, 13, 14, 15 are included.
Normal Approximation to Binomial Example 1 (continued)
Normal Approximation to Binomial Continuity Correction
To make proper continuity correction, list # of successes included in the event The continuity correction allows us to calculate P(X=x)
Binomial Probability P(21 < X 25) P(X = 10) P(5 X < 12) No. of Successes Normal Curve w/CC
Normal Approximation to Binomial Example 2
Consider a lottery in which you have a ten percent chance of winning. Suppose you play 10,000 times. What is the probability of winning at least 1030 times?
Normal Approximation to Binomial Example 2 (continued)
Using Excel to Compute Standard Normal Probabilities
Excel has two functions for computing probabilities and z values for a standard normal distribution
NORMSDIST is used to compute the cumulative probability given a value of z NORMSINV is used to compute the z value given a cumulative probability
Using Excel to Compute Standard Normal Probabilities
Excel Formula Worksheet
1 2 3 4 5 6 7 8 9 A B Probabilities: Standard Normal Distribution P (z < 1.00) P (0.00 < z < 1.00) P (0.00 < z < 1.25) P (-1.00 < z < 1.00) P (z > 1.58) P (z < -0.50) =NORMSDIST(1) =NORMSDIST(1)-NORMSDIST(0) =NORMSDIST(1.25)-NORMSDIST(0) =NORMSDIST(1)-NORMSDIST(-1) =1-NORMSDIST(1.58) =NORMSDIST(-0.5)
Using Excel to Compute Standard Normal Probabilities
Excel Value Worksheet
1 2 3 4 5 6 7 8 9 A B Probabilities: Standard Normal Distribution P (z < 1.00) P (0.00 < z < 1.00) P (0.00 < z < 1.25) P (-1.00 < z < 1.00) P (z > 1.58) P (z < -0.50) 0.8413 0.3413 0.3944 0.6827 0.0571 0.3085
Using Excel to Compute Standard Normal Probabilities
Excel Formula Worksheet
1 2 3 4 5 6 A B Finding z Values, Given Probabilities z value with .10 in upper tail z value with .025 in upper tail z value with .025 in lower tail =NORMSINV(0.9) =NORMSINV(0.975) =NORMSINV(0.025)
Using Excel to Compute Standard Normal Probabilities
Excel Value Worksheet
1 2 3 4 5 6 A B Finding z Values, Given Probabilities z value with .10 in upper tail z value with .025 in upper tail z value with .025 in lower tail 1.28 1.96 -1.96
Using Excel to Compute Normal Probabilities
Excel has two functions for computing cumulative probabilities and x values for any normal distribution: NORMDIST is used to compute the cumulative probability given an x value NORMINV is used to compute the x value given a cumulative probability.
Using Excel to Compute Normal Probabilities
Excel Formula Worksheet
1 2 3 4 5 6 7 8 A B Probabilities: Normal Distribution P (x > 20) =1-NORMDIST(20,15,6,TRUE) Finding x Values, Given Probabilities x value with .05 in upper tail =NORMINV(0.95,15,6)
Using Excel to Compute Normal Probabilities
Excel Value Worksheet
1 2 3 4 5 6 7 8 A B Probabilities: Normal Distribution P (x > 20) 0.2023 Finding x Values, Given Probabilities x value with .05 in upper tail 24.87
Note: P(x > 20) = .2023 here using Excel, while our previous manual approach using the z table yielded .2033 due to our rounding of the z value.