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An introduction to probability theory and its applications by William Feller

By William Feller

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What is the value of α, approximately? 6 The Weibull distribution is a generalization of the exponential distribution that comes up in failure analysis (see ❤tt♣✿✴✴✇✐❦✐♣❡❞✐❛✳♦r❣✴ ✇✐❦✐✴❲❡✐❜✉❧❧❴❞✐str✐❜✉t✐♦♥). Its CDF is CDF ( x ) = 1 − e−( x/λ) k Can you nd a transformation that makes a Weibull distribution look like a straight line? What do the slope and intercept of the line indicate? Use r❛♥❞♦♠✳✇❡✐❜✉❧❧✈❛r✐❛t❡ to generate a sample from a Weibull distribution and use it to test your transformation.

If they are big enough to smooth out noise, they might also smooth out useful information. An alternative that avoids these problems is the cumulative distribution function, or CDF. But before we can get to that, we have to talk about percentiles. 3 Percentiles If you have taken a standardized test, you probably got your results in the form of a raw score and a percentile rank. In this context, the percentile rank is the fraction of people who scored lower than you (or the same). So if you are “in the 90th percentile,” you did as well as or better than 90% of the people who took the exam.

7 Plotting PMFs There are two common ways to plot Pmfs: • To plot a Pmf as a bar graph, you can use ♣②♣❧♦t✳❜❛r or ♠②♣❧♦t✳❍✐st. Bar graphs are most useful if the number of values in the Pmf is small. • To plot a Pmf as a line, you can use ♣②♣❧♦t✳♣❧♦t or ♠②♣❧♦t✳P♠❢. Line plots are most useful if there are a large number of values and the Pmf is smooth. 2 shows the PMF of pregnancy lengths as a bar graph. Using the PMF, we can see more clearly where the distributions differ. 8. 2: PMF of pregnancy lengths.

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