1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.

Lecture 6

1Outline

1. Random Variablesa. Discrete Random Variables

b. Continuous Random Variables

2. Symmetric Distributions

3. Normal Distributions

4. The Standard Normal Distribution

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21. Random Variables

Two kinds of random variables:

a. Discrete (DRV) Outcomes have countable values Possible values can be listed E.g., # of people in this room

Possible values can be listed: might be …28 or 29 or 30…

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31. Random Variables

Two kinds of random variables:

b. Continuous (CRV) Not countable Consists of points in an interval E.g., time till coffee break

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41. Random Variables

The form of the probability distribution for a CRV is a smooth curve. Such a distribution may also be called a

Frequency Distribution Probability Density Function

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51. Random Variables

In the graph of a CRV, the X axis is whatever you are measuring (e.g., exam scores, depression scores, # of widgets produced per hour).

The Y axis measures the frequency of scores.

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X

The Y-axis measures frequency. It is usually not shown.

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72. Symmetric Distributions

In a symmetric CRV, 50% of the area under the curve is in each half of the distribution.

P(x ≤ ) = P(x ≥ ) = .5

Note: Because points are infinitely thin, we can only measure the probability of intervals of X values – not of individual X values.

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8

µ

50% of area

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93. Normal Distributions

A particularly important set of CRVs have probability distributions of a particular shape: mound-shaped and symmetric. These are “normal distributions”

Many naturally-occurring variables are normally distributed.

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10Normal Distributions

are perfectly symmetrical around their mean, .

have the standard deviation, , which measures the “spread” of a distribution – an index of variability around the mean.

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11

µ

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12Standard Normal Distribution

The area under the curve between and some value X ≥ has been calculated for the “standard normal distribution” and is given in the Z table (Table IV).

E.g., for Z = 1.62, area = .4474

(Note that for the mean, Z = 0.)

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XZ = 1.62Z = 0

Area gives the probability of finding a score between the mean and X when you make an observation

.4474

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14Using the Standard Normal Distribution

Suppose average height for Canadian women is 160 cm, with = 15 cm.

What is the probability that the next Canadian woman we meet is more than 175 cm tall?

Note that this is a question about a single case and that it specifies an interval.

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15Using the Standard Normal Distribution

160 175

We need this areaTable gives this area

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Remember that area above the mean, , is half (.5) of the distribution.

µ

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17Using the Standard Normal Distribution

160 175

Call this shaded area P. We can get P from Table IV

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18Using the Standard Normal Distribution

Z = X - = 175-160

15

= 1.00

Now, look up Z = 1.00 in the table.

Corresponding area (= probability) is P = .3413.

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19Using the Standard Normal Distribution

160 175

This area is .3413

So this area must be .5 – .3413 = .1587

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20Using the Standard Normal Distribution

Z = 0 Z = 1.0

This area is .3413

So this area must be .5 – .3413 = .1587

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21Using the Standard Normal Distribution

What is the probability that the next Canadian woman we meet is more than 175 cm tall?

Answer: .1587

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22Review

Area under curve gives probability of finding X in a given interval. Area under the curve for Standard Normal Distribution is given in Table IV. For area under the curve for other normally-distributed variables first compute:

Z = X -

Then look up Z in Table IV.