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Lecture 8

Lecture 8. Normal Distribution and Standard Normal Distribution. The Normal Distribution The most important continuous distribution. The continuous rv X, whose pdf is given above, is said to be normally distributed with parameters  and  2 . We often use the abbreviation N(, 2 ).

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Lecture 8

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  1. Lecture 8 Normal Distribution and Standard Normal Distribution

  2. The Normal Distribution The most important continuous distribution. The continuous rv X, whose pdf is given above, is said to be normally distributed with parameters  and 2. We often use the abbreviation N(,2). Time it takes to accomplish a task Weight, height of people, objects Dimensions of products Test scores Economic measures, or indicators, etc. Are examples of data that often are best described by Normal Distribution

  3. Properties of the normal distribution • E(X) = , and V(X) = 2 • Symmetric about  and bell-shaped. • Center of the bell (peak) is both the mean and the median of the distribution. • The value of  is the distance to the inflection points on the curve. a- N(0,1), =0, 2=1, =1 b- N(0,4), =0, 2=4, =2 c- N(3,4), =4, 2=4, =2 a c b

  4. Percentages under the normal curve (Percentage vales are approximations)

  5. The Standard Normal Distribution To compute any P(a  x  b) for a normal distribution the following integral needs to be computed: However, it is analytically not possible to compute the above integral. With today’s computers we can numerically compute these integrals but tabulated information can be found about the standard normal curve (=0, =1) which can be used to compute the probabilities for any value of  and .

  6. By definition, the standard normal distribution is N(0,1). We denote the standard normal random variable by Z. • The cdf of Z is P(Zz) = • The cdf at z is denoted by (z) = The probability that Z  z Turn to page# Area= (z) 0 z

  7. Examples • Lets compute the following standard normal probabilities: • (a) P(Z1.25), (b) P(Z>1.25), (c) P(Z-1.25), • (d) P(-0.38 Z1.25) • (a) From page 741, (1.25) = 0.8944 • (b) P(Z>1.25) = 1  P(Z1.25) = 1  0.8944 = 0.1056 • P(Z>1.25) = P(Z1.25) = 0.1056 • (c) P(Z-1.25) => We can use the table on page 740, but some tables contain only positive z values. From symmetry; • P(Z-1.25) = P(Z1.25) = 0.1056 • (d) P(-0.38  Z  1.25) = (1.25)  (-0.38) = 0.8944 – 0.3520 P(-0.38  Z  1.25) = 0.5424

  8. Percentiles of the standard normal curve • As we all know by now, the 99th percentile is the value of the rv, less than which we expect to see the 99 percent of the entire population. • This is the inverse problem of calculating the probability. Instead of P(Z  z)=?, now we consider: P(Z  ?) = 0.99 (example for the 99th percentile) P(Z  ?) = 0.25 (for the 25th percentile) etc. • Z notation:  denotes the area under the Z curve that lies to the right of z. • Z0.01 = the 99th percentile. 0.99 99th percentile 0.99 Area = 0.01 Z0.01

  9. The rv Z has a standard normal distribution. Therefore, we can use the following expressions to calculate the probabilities associated with X: • Non-standard Normal Distributions We standardize a normal rv by:

  10. Example: Example 4.15 on page 166 Time to react to the brake light X~N(1.25,0.462) P(1.00  X  1.75) = ?

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