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Lesson 5:

Lesson 5:. Continuous Probability Distributions. Outline. Types of Probability Distributions. Probability distribution may be classified according to the number of random variables it describes. Continuous Probability Distributions.

erin-osborn
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Lesson 5:

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  1. Lesson 5: Continuous Probability Distributions

  2. Outline

  3. Types of Probability Distributions Probability distribution may be classified according to the number of random variables it describes.

  4. Continuous Probability Distributions The curve f(x) is the continuous probability distribution (or probability curve or probability density function) of the random variable X if the probability that X will be in a specified interval of numbers is the area under the curve f(x) corresponding to the interval. Properties of f(x) • f(x)  0 for all x • The total area under the curve of f(x) is equal to 1

  5. Features of a Univariate Continuous Distribution • Let X be a random variable that takes any real values in an interval between a and b. The number of possible outcomes are by definition infinite. • The main features of a probability density function f(x) are: • f(x)  0 for all x and may be larger than 1. • The probability that X falls into an subinterval (c,d) is and lies between 0 and 1. • P(X  (a,b)) = 1. • P(X = x) = 0.

  6. The Univariate Uniform Distribution If c and d are numbers on the real line, the random variable X ~ U(c,d), i.e., has a univariate uniform distribution if The mean and standard deviation of a uniform random variable x are

  7. The Uniform Density

  8. The Normal Probability Distribution The random variable X ~ N(,2), i.e., has a univariate normal distribution if for all x on the real line (-,+ ) • and  are the mean and standard deviation,  = 3.14159 … and e = 2.71828 is the base of natural or Naperian logarithms.

  9. Learning exercise 4: Part-time Work on Campus • A student has been offered part-time work in a laboratory. The professor says that the work will vary from week to week. The number of hours will be between 10 and 20 with a uniform probability density function, represented as follows: • How tall is the rectangle? • What is the probability of getting less than 15 hours in a week? • Given that the student gets at least 15 hours in a week, what is the probability that more than 17.5 hours will be available?

  10. Learning exercise 4: Part-time Work on Campus • How tall is the rectangle? • (20-10)*h = 1 • h=0.1 • What is the probability of getting less than 15 hours in a week? • 0.1*(15-10) = 0.5 • Given that the student gets at least 15 hours in a week, what is the probability that more than 17.5 hours will be available? • 0.1*(20-17.5) = 0.25 • 0.25/0.5 = 0.5 P(hour>17.5)/P(hour>15)

  11. Features of a Bivariate Continuous Distribution • Let X1 and X2 be a random variables that takes any real values in a region (rectangle) of (a,b,c,d). The number of possible outcomes are by definition infinite. • The main features of a probability density function f(x1,x2) are: • f(x1,x2)  0 for all (x1,x2) and may be larger than 1. • The probability that (X1,X2) falls into a region (rectangle) or (p,q,r,s) is and lies between 0 and 1. • P((X1,X2)  (a,b,c,d)) = 1. • P((X1,X2) = (x1,x2)) = 0.

  12. The Bivariate Uniform Distribution If a, b, c and d are numbers on the real line, , the random variable (X1,X2) ~ U(a,b,c,d), i.e., has a bivariate uniform distribution if

  13. The Marginal Density • The marginal density functions are:

  14. The Conditional Density • The conditional density functions are:

  15. The Expectation (Mean) of Continuous Probability Distribution • For univariate probability distribution, the expectation or mean E(X) is computed by the formula: • For bivariate probability distribution, the the expectation or mean E(X) is computed by the formula:

  16. Conditional Mean of Bivariate Discrete Probability Distribution • For bivariate probability distribution, the conditional expectation or conditional mean E(X|Y) is computed by the formula: • Unconditional expectation or mean of X, E(X)

  17. Expectation of a linear transformed random variable • If a and b are constants and X is a random variable, then E(a) = a E(bX) = bE(X) E(a+bX) = a+bE(X)

  18. The Variance of a Continuous Probability Distribution • For univariate continuous probability distribution • If a and b are constants and X is a random variable, then V(a) = 0 V(bX) = b2V(X) V(a+bX) = b2V(X)

  19. The Covariance of a Bivariate Discrete Probability Distribution Covariance measures how two random variables co-vary. • If a and b are constants and X is a random variable, then C(a,b) = 0 C(a,bX) = 0 C(a+bX,Y) = bC(X,Y)

  20. Variance of a sum of random variables • If a and b are constants and X and Y are random variables, then V(X+Y) = V(X) + V(Y) + 2C(X,Y) V(aX+bY) =a2V(X) + b2V(Y) + 2abC(X,Y)

  21. Correlation coefficient • The strength of the dependence between X and Y is measured by the correlation coefficient:

  22. Importance of Normal Distribution • Describes many random processes or continuous phenomena • Basis for Statistical Inference

  23. Characteristics of a Normal Probability Distribution • bell-shaped and single-peaked (unimodal) at the exact center of the distribution.

  24. Characteristics of a Normal Probability Distribution • Symmetrical about its mean. The arithmetic mean, median, and mode of the distribution are equal and located at the peak. Thus half the area under the curve is above the mean and half is below it.

  25. Characteristics of a Normal Probability Distribution • The normal probability distribution is asymptotic. That is the curve gets closer and closer to the X-axis but never actually touches it.

  26. N(0,2) Bell-shaped Symmetric Mean=median = mode Unimodal Asymptotic

  27. N(,2) x (a) x (b) x (c)

  28. Normal Distribution Probability Probability is the area under the curve! A table may be constructed to help us find the probability f(X) X d c

  29. Infinite Number of Normal Distribution Tables Normal distributions differ by mean & standard deviation. f(X) Each distribution would require its own table. X

  30. The Standard Normal Probability Distribution -- N(0,1) • The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. • It is also called the z distribution. • A z-value is the distance between a selected value, designated X, and the population mean , divided by the population standard deviation, . The formula is:

  31. Transform to Standard Normal Distribution -- A numerical example • Any normal random variable can be transformed to a standard normal random variable

  32. The Standard Normal Probability Distribution • Any normal random variable can be transformed to a standard normal random variable • Suppose X ~ N(µ, 2) • Z=(X-µ)/  ~ N(0,1) • P(X<k) = P [(X-µ)/  < (k-µ)/  ]

  33. Standardize the Normal Distribution Normal Distribution Standardized Normal Distribution = 1   z   = 0 X Z Z Because we can transform any normal random variable into standard normal random variable, we need only one table!

  34. Standardizing Example Normal Distribution Standardized Normal Distribution  = 10  = 1 Z = 5   = 0 .12 Z X 6.2 Z

  35. Obtaining the Probability Standardized Normal Probability Table (Portion) .02 Z .00 .01  = 1 Z .0000 0.0 .0040 .0080 0.0478 .0398 .0438 0.1 .0478 0.2 .0793 .0832 .0871  = 0 0.12 Z Z 0.3 .1179 .1217 .1255 Shaded Area Exaggerated Probabilities

  36. Example P(3.8 X 5) Normal Distribution Standardized Normal Distribution  = 10  = 1 Z 0.0478 = 5  = 0 Z 3.8  X -0.12 Z Shaded Area Exaggerated

  37. Example (2.9 X 7.1) X   2 . 9  5 Z     . 21  10 X   7 . 1  5 Normal Distribution Standardized Normal Distribution Z    . 21  10  = 10  = 1 Z .1664 .0832 .0832 Z 2.9 5 7.1 X -.21 0 .21 Shaded Area Exaggerated

  38. Example (2.9 X 7.1) X   2 . 9  5 Z     . 21  10 X   7 . 1  5 Normal Distribution Standardized Normal Distribution Z    . 21  10  = 10  = 1 Z .1664 .0832 .0832 Z 2.9 5 7.1 X -.21 0 .21 Shaded Area Exaggerated

  39. Example P(X 8) X   8  5 Z    . 30 Normal Distribution Standardized Normal Distribution  10  = 10  = 1 Z .5000 .3821 .1179  = 0 .30 Z  = 5 8 X Z Shaded Area Exaggerated

  40. Example P(7.1 X 8)   7 . 1  5 X Z    . 21  10 X   8  5 Normal Distribution Z    . 30 Standardized Normal Distribution  10  = 10  = 1 Z .1179 .0347 .0832 Z  = 5 X  = 0 7.1 8 .21 .30 z Shaded Area Exaggerated

  41. Normal Distribution Thinking Challenge • You work in Quality Control for GE. Light bulb life has a normal distribution with µ= 2000 hours &  = 200 hours. What’s the probability that a bulb will last • between 2000 & 2400 hours? • less than 1470 hours?

  42. Solution P(2000 X 2400) P(2000<X<2400) = P [(2000-µ)/  <(X-µ)/  < (2400-µ)/  ] = P[(X-µ)/  < (2400-µ)/  ] – P [(X-µ)/  < (2000-µ)/  ] = P[(X-µ)/  < (2400-µ)/  ] – 0.5 Normal Distribution Standardized Normal Distribution X   2400  2000 Z    2 . 0  200  = 200  = 1 Z .4772 Z X  = 0 2.0  = 2000 2400 Z Shaded Area Exaggerated

  43. Solution P(X 1470) P(X<1470) = P [(X-µ)/  < (1470-µ)/  ] X   1470  2000 Z     2 . 65  200 Normal Distribution Standardized Normal Distribution  = 200  = 1 Z .5000 .4960 .0040 Z X -2.65  = 0 1470  = 2000 Z Shaded Area Exaggerated

  44. Finding Z Values for Known Probabilities What Is Z Given P(Z) = 0.1217? Standardized Normal Probability Table (Portion) .01  = 1 Z .00 .02 Z .1217 0.0 .0040 .0080 .0000 0.1 .0398 .0438 .0478 0.2 .0793 .0832 .0871  = 0 .31 Z Z .1179 .1255 0.3 .1217 Shaded Area Exaggerated

  45. Finding X Values for Known Probabilities Normal Distribution Standardized Normal Distribution  = 10  = 1 Z .1217 .1217 ?  = 0 .31 Z  = 5 X Z Shaded Area Exaggerated

  46. EXAMPLE 1 • The bi-monthly starting salaries of recent MBA graduates follows the normal distribution with a mean of $2,000 and a standard deviation of $200. What is the z-value for a salary of $2,400?

  47. EXAMPLE 1 continued • What is the z-value of $1,900 ? • Az-value of 2 indicates that the value of $2,400 is one standard deviation above the mean of $2,000. • A z-value of –1.50 indicates that $1,900 is 1.5 standard deviation below the mean of $2000.

  48. Areas Under the Normal Curve • About 68 percent of the area under the normal curve is within one standard deviation of the mean. ± • P(-  < X < + ) = 0.6826 • About 95 percent is within two standard deviations of the mean. ± 2  • P(- 2  < X < + 2 ) = 0.9544 • Practically all is within three standard deviations of the mean. ± 3  • P(- 3  < X < + 3 ) = 0.9974

  49. EXAMPLE 2 • The daily water usage per person in New Providence, New Jersey is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons. • About 68 percent of those living in New Providence will use how many gallons of water? • About 68% of the daily water usage will lie between 15 and 25 gallons.

  50. EXAMPLE 2 continued • What is the probability that a person from New Providence selected at random will use between 20 and 24 gallons per day? P(20<X<24) =P[(20-20)/5 < (X-20)/5 < (24-20)/5 ] =P[ 0<Z<0.8 ] The area under a normal curve between a z-value of 0 and a z-value of 0.80 is 0.2881. We conclude that 28.81 percent of the residents use between 20 and 24 gallons of water per day.

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