11. Tilted distributions. (a) Let X have distribution function F and let T be such that M(T) = E(e¹x) < <∞o. Show that F(x) = M(t)-¹ Sety dF (y) is a distribution function, called a 'tilted distribution' of X, and find its moment generating function. (b) Suppose X and Y are independent and E(et X), E(e¹Y)<∞o. Find the moment generating function of the tilted distribution of X + Y in terms of those of X and Y.

11. Tilted distributions. (a) Let X have distribution function F and let T be such that M(T) = E(e¹x) < <∞o. Show that F(x) = M(t)-¹ Sety dF (y) is a distribution function, called a 'tilted distribution' of X, and find its moment generating function. (b) Suppose X and Y are independent and E(et X), E(e¹Y)<∞o. Find the moment generating function of the tilted distribution of X + Y in terms of those of X and Y.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter2: Exponential, Logarithmic, And Trigonometric Functions

Section2.CR: Chapter 2 Review

Problem 111CR: Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data...

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Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.
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