121616949-math.228 - ce-cx x ≥ where c is a positive...

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214 Chapter 9 Applications of Integration is finite and positive. If we let g ( x ) = f ( x ) /A , then g is a probability density function. It turns out to be very useful, and is called the standard normal probability den- sity function or more informally the bell curve , giving rise to the standard normal distribution . See figure 9.17 for the graph of the bell curve. - 4 - 3 - 2 - 1 0 1 2 3 4 0.5 . Figure 9.17 The bell curve. We have shown that A is some finite number without computing it; we cannot compute it with the techniques we have available. By using some techniques from multivariable calculus, we can show that A = 2 π . EXAMPLE 9.27 The exponential distribution has probability density function f ( x ) = n 0 x < 0 ce - cx x 0 where c is a positive constant. DEFINITION 9.28 Let f be a probability density function. Since we are assuming
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Unformatted text preview: ce-cx x ≥ where c is a positive constant. DEFINITION 9.28 Let f be a probability density function. Since we are assuming that the function F ( x ) = Z x-∞ f ( t ) dt is continuous and ranges from 0 to 1 it follows by the intermediate value theorem that there is at least one number m such that F ( m ) = 1 / 2; m is called a median of the random variable X . Note that P ( X ≤ m ) = P ( X ≥ m ) = 1 / 2. EXAMPLE 9.29 The median of the uniform distribution on [ a, b ] is ( a + b ) / 2. EXAMPLE 9.30 Let f ( x ) = x < 1 ≤ x ≤ 1 / 2 1 / 2 < x < 1 1 1 ≤ x ≤ 3 / 2 3 / 2 < x . Any value in the interval [1 / 2 , 1 qualifies as a median, although convention would usually take ‘the’ median to be 3 / 4....
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