CONTINUOUS RANDOM VARIABLES AND THE NORMAL DISTRIBUTION

2 CONTINUOUS PROBABILITY DISTRIBUTION Two characteristics 1. The probability that x assumes a value in any interval lies in the range 0 to 1 2. The total probability of all the intervals within which x can assume a value of 1.0

3 Area under a curve between two points. x = a x = b x Shaded area is between 0 and 1

4 Total area under a probability distribution curve. Shaded area is 1.0 or 100% x

5 Area under the curve as probability. a b x Shaded area gives the probability P ( a ≤ x ≤ b )

6 The probability of a single value of x is zero.

7 THE NORMAL DISTRIBUTION  Normal Probability Distribution  A normal probability distribution, when plotted, gives a bell-shaped curve such that 1. The total area under the curve is The curve is symmetric about the mean. 3. The two tails of the curve extend indefinitely.

8 Normal distribution with mean μ and standard deviation σ. Standard deviation = σ Mean = μ x

9 Total area under a normal curve. The shaded area is 1.0 or 100% μx

10 A normal curve is symmetric about the mean. Each of the two shaded areas is.5 or 50%.5 μx

11 Areas of the normal curve beyond μ ± 3σ. μ – 3σ μ – 3σ μ + 3 σ Each of the two shaded areas is very close to zero μ x

12 Three normal distribution curves with the same mean but different standard deviations. σ = 5 σ = 10 σ = 16 x μ = 50

13 Three normal distribution curves with different means but the same standard deviation. σ = 5 σ = 5 σ = 5 µ = 20 µ = 30 µ = 40 x

14 THE STANDARD NORMAL DISTRIBTUION  Definition  The normal distribution with μ = 0 and σ = 1 is called the standard normal distribution.

15 The standard normal distribution curve. σ = 1 µ = z

16 THE STANDARD NORMAL DISTRIBTUION  Definition  The units marked on the horizontal axis of the standard normal curve are denoted by z and are called the z values or z scores. A specific value of z gives the distance between the mean and the point represented by z in terms of the standard deviation.

17 Area under the standard normal curve z. 5 Each of these two areas is.5

18 Examples Using The Standard Normal Table …

19 STANDARDIZING A NORMAL DISTRIBUTION  Converting an x Value to a z Value  For a normal random variable x, a particular value of x can be converted to its corresponding z value by using the formula  where μ and σ are the mean and standard deviation of the normal distribution of x, respectively.

20 Examples … IQ ~ Normal mean = 100 stdev = 15 …

21 DETERMINING THE z AND x VALUES WHEN AN AREA UNDER THE NORMAL DISTRIBUTION CURVE IS KNOWN Examples …

22 Finding an x Value for a Normal Distribution For a normal curve, with known values of μ and σ and for a given area under the curve the x value is calculated as x = μ + zσ