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(a) Explain why ri has unit standard deviation. (b) Do the standardized residuals have unit standard deviation (c) Discuss the behavior of the studentized residual when the sample value xi is very close to the middle of the range of x. (d) Discuss the behavior of the studentized residual when the sample value xi is very near one end of the range of x.

... 6+V+K:3 1 ...

400 11-9

We occasionally nd that the straight-line regression model Y is inappropri0 1x ate because the true regression function is nonlinear. Sometimes nonlinearity is visually determined from the scatter diagram, and sometimes, because of prior experience or underlying theory, we know in advance that the model is nonlinear. Occasionally, a scatter diagram will exhibit an apparent nonlinear relationship between Y and x. In some of these situations, a nonlinear function can be expressed as a straight line by using a suitable transformation. Such nonlinear models are called intrinsically linear. As an example of a nonlinear model that is intrinsically linear, consider the exponential function Y

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Reduces contrast and softly mutes the colors of your scene Ideal for bright sunny days with lots of contrast or when you want a less intense look to the colors in a scene The opposite of Vivid Sepia Converts your scene to black and white and adds a warm brown, nostalgic tint to the image B/W De-saturates your image colors completely and creates a black-and-white or monotone image Positive Film Mixes the effects of Vivid Red, Vivid Green, and Vivid Blue to create realistic intense colors reminiscent of high-saturation slide (positive) lms like Fujichrome and Ecktachrome If you miss the look of your old slide lms when shooting digitally, you can change the images you shoot to this look Lighter Skin Tone.

This function is intrinsically linear, since it can be transformed to a straight line by a logarithmic transformation ln Y ln

assuming that, after reduction modulo 256, we have 6 V K3 > 5 V . If this is not the case, then 6 V K3 will appear to the left of 5 V ,but this has no effect on the success of the attack. Now suppose for a moment that the RC4 initialization algorithm were to stop after the i = 3 step. Then if we generate the first byte of the keystream according to the algorithm in Table 3.10, we find i = 1 and j = Si = S = 0, 1 so that t = S SO= 0 3 = 3. Then the first keystream byte would be 1

0 1x

This transformation requires that the transformed error terms ln are normally and independently distributed with mean 0 and variance 2. Another intrinsically linear function is Y By using the reciprocal transformation z Y

+ K3) (mod 256).

0 1z

Sometimes several transformations can be employed jointly to linearize a function. For example, consider the function Y letting Y* exp 1 1

(3.10)

1 Y , we have the linearized form ln Y*

0 1x

For examples of tting these models, refer to Montgomery, Peck, and Vining (2001) or Myers (1990).

Assuming that Trudy knows (or can guess) the first byte of the plaintext, she can determine the first byte of the keystream. If this is the case, Trudy can simply solve (3.10) to obtain the first unknown key byte, since

11-10 11-11

Lightens up most skin tones to make them appear brighter, but may have trouble with certain skin types and can also affect other areas of a scene besides skin Experiment to see the differences for yourself Darker Skin Tone Darkens skin tones slightly to produce a normal skin tone look with subjects that have pale or washed-out skin Can make certain skin tones too dark and may not provide the desired effect, and can also affect other areas of the image besides skin Vivid Blue Punches up the blues in your photographs Works great on oceans, skies, and any other blue objects in the photo by adding saturation and intensity to only the blue areas in the image Vivid Green Intensi es the green areas in your photograph such as grass, foliage, mountains, and any other green objects included in the frame.

Our development of regression analysis has assumed that x is a mathematical variable, measured with negligible error, and that Y is a random variable. Many applications of regression analysis involve situations in which both X and Y are random variables. In these situations, it

K3 =

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