Normal Distribution. Distribution function and Graphical Representation - pdf - identifying the mean and variance

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4 Distribution function and Graphical Representation - pdf - identifying the mean and variance f ( x ) 1 ( ) x e

5 Distribution function and Graphical Representation - pdf - identifying the mean and variance f ( x) 1 e 8 ( x7) 8

6 Distribution function and Graphical Representation - pdf - identifying the mean and variance f ( x) 1 e 8 ( x7) 8 X ~ N(7,4)

7 Distribution function and Graphical Representation - pdf - identifying the mean and variance 4 ) ( 1 ) ( x e x f ) ( 1 ) ( x e x f

8 Distribution function and Graphical Representation - pdf - identifying the mean and variance ) ( 1 ) ( x e x f ) ( 1 ) ( x e x f X ~ N(,)

9 Distribution function and Graphical Representation - pdf - identifying the mean and variance - standard normal f ( x ) 1 ( ) x e

10 Distribution function and Graphical Representation - pdf - identifying the mean and variance - standard normal 1 ) ( x e x f ) ( 1 ) ( x e x f

11 Distribution function and Graphical Representation - symmetry - effect of mean and variance changes

12 Calculations general concepts of determining probabilities usefulness of tables

13 Calculations general concepts of determining probabilities usefulness of tables taking advantage of symmetry

14 Examples Suppose X ~ N(0,1). Calculate the following: a) P(X 0)

15 Examples Suppose X ~ N(0,1). Calculate the following: a) P(X 0) b) P(X 1.3)

16 Examples Suppose X ~ N(0,1). Calculate the following: a) P(X 0) b) P(X 1.3) c) P(X 1.3)

17 Examples Suppose X ~ N(0,1). Calculate the following: d) P( 0.83 X 0.75)

18 Examples Suppose X ~ N(0,1). Calculate the following: d) P( 0.83 X 0.75) e) P( X 3)

19 Examples Suppose X ~ N(0,1). Calculate the following: d) P( 0.83 X 0.75) e) P( X 3) f) P(X 1.735)

20 What about X ~ N(10,4)?

21 What about X ~ N(10,4)? Want to shift the graph to a mean of zero, and decrease the standard deviation to 1.

22 What about X ~ N(10,4)? Want to shift the graph to a mean of zero, and decrease the standard deviation to 1. Define a new random variable Z = X 10 Z ~ N(0,1)

23 What about X ~ N(10,4)? Want to shift the graph to a mean of zero, and decrease the standard deviation to 1. Define a new random variable Z = X 10 Z ~ N(0,1) E[Z]? Var[Z]?

24 What about X ~ N(10,4)? Want to shift the graph to a mean of zero, and decrease the standard deviation to 1. Define a new random variable Z = X 10 Z ~ N(0,1) Standardization In general, X ~ N(, ) Z = X ~ N(0,1)

25 What about X ~ N(10,4)? Want to shift the graph to a mean of zero, and decrease the standard deviation to 1. Define a new random variable Z = X 10 Z ~ N(0,1) Standardization In general, X ~ N(, ) Z = X ~ N(0,1) After standardization, calculation concepts are the same as before.

26 Example Suppose X ~ N(3,4). Calculate P(X 6).

27 Review Questions Suppose that X is a normally distributed random variable. Calculate P(4.5 < X < 6.5) where E(X) = 5 and Standard Deviation of X = None of these

28 Review Questions Suppose that a population of men's heights is normally distributed with a mean of 68 inches, and variance of 9 inches. What s the probability of a randomly chosen man being taller than 6 feet? None of these

29 Example Suppose that the height of a tomato plant can be approximated as a normal random variable with a mean of 9.4cm and a standard deviation of.1cm. Calculate the probability of growing a plant between 9 and 30cm tall.

30 Example (#35) A Wall street analyst estimates that the annual return from the stock of Company A can be considered to be an observation from a normal distribution with mean µ= 8% and a standard deviation =1.5%. The analyst s investment choices are based upon the consideration that any return greater than 5% is satisfactory, and a return greater than 10% is excellent.

31 Example (#35) A Wall street analyst estimates that the annual return from the stock of Company A can be considered to be an observation from a normal distribution with mean µ= = 8% and a standard deviation =1.5%. The analyst s investment choices are based upon the consideration that any return greater than 5% is satisfactory, and a return greater than 10% is excellent.

32 Example (#35) A Wall street analyst estimates that the annual return from the stock of Company A can be considered to be an observation from a normal distribution with mean µ= = 8% and a standard deviation =1.5%. The analyst s investment choices are based upon the consideration that any return greater than 5% is satisfactory, and a return greater than 10% is excellent. Calculate the probability that Company A s stock will prove to be: a) Unsatisfactory b) Excellent

33 Using the table backwards Suppose that Z N(0,1). Find the value of x for which P(Z x) = 0.85

34 Using the table backwards Suppose that Z N(0,1). Find the value of x for which P(Z x) = 0.85

35 Using the table backwards Suppose that Z N(0,1). Find the value of x for which P(Z x) = 0.85 x 1.035

36 Using the table backwards Suppose that Z N(0,1). Find the value of x for which P(Z x) = 0.3

37 Using the table backwards Suppose that Z N(0,1). Find the value of x for which P(Z x) = 0.3 Positive side starts at 0.50

38 Using the table backwards Suppose that Z N(0,1). Find the value of x for which P(Z x) = 0.3 x y

39 Using the table backwards Suppose that Z N(0,1). Find the value of x for which P(Z x) = 0.3 Find y such that (y) = 0.77 y = x x y

40 Using the table backwards Suppose that Z N(0,1). Find the value of x for which P(Z x) = 0.3 Find y such that (y) = 0.77 y = x x = x y

41 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75]

42 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75] P(X x) = not standard Normal

43 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75] P(X x) = P( X 7 x 7 )

44 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75] P(X x) = P( X 7 x 7 ) = P( Z x 7 ) = 0.75

45 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75] P(X x) = P( X 7 x 7 ) = P( Z x 7 ) = 0.75

46 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75] P(X x) = P( X 7 x 7 ) = P( Z x 7 ) = 0.75 x

47 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75] P(X x) = P( X 7 x 7 ) = P( Z x 7 ) = 0.75 x 8.35

48 Using the table backwards Determining x values when given probabilities Determining percentiles

49 Using the table backwards Determining x values when given probabilities Determining percentiles Critical points, z P(X z ) = z

50 Using the table backwards Determining x values when given probabilities Determining percentiles Critical points, z P(X z ) = 1 e.g., z 0.1 = z

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52 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75] P(X x) = P( X 7 x 7 ) = P( Z x 7 ) = 0.75

53 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75] P(X x) = P( X 7 x 7 ) = P( Z x 7 ) = 0.75 x

54 Using the table backwards Suppose that X N(7,4). Find the upper quartile [the value of x for which (x) = 0.75] P(X x) = P( X 7 x 7 ) = P( Z x 7 ) = 0.75 x 8.35

55 Using the table backwards Determining x values when given probabilities Determining percentiles

56 Using the table backwards Determining x values when given probabilities Determining percentiles Critical points, z P(X z ) = z

57 Using the table backwards Determining x values when given probabilities Determining percentiles Critical points, z P(X z ) = 1 e.g., z 0.1 = z