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Suppose the random variable T is the length of life of an object (possibly the lifetime of an electrical component or of a subject given a particular treatment). The hazard function hr(t) associated with the random variable T is defined by hr(t) = lims-o- P(t ≤T t) 8 Thus, we can interpret hr(t) as the rate of change of the probability that the object survives a little past time t, given that the object survives to time t. Show that if T is a continuous random variable, then hr(t) - fr(t) 1- Fr(t) = d dt log (1 - Fr(t)).

Suppose the random variable T is the length of life of an object (possibly the lifetime of an electrical component or of a subject given a particular treatment). The hazard function hr(t) associated with the random variable T is defined by hr(t) = lims-o- P(t ≤T t) 8 Thus, we can interpret hr(t) as the rate of change of the probability that the object survives a little past time t, given that the object survives to time t. Show that if T is a continuous random variable, then hr(t) - fr(t) 1- Fr(t) = d dt log (1 - Fr(t)).

Calculus For The Life Sciences
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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Suppose the random variable T is the length of life of an object (possibly the lifetime of an electrical component or of a subject given a particular treatment). The hazard function hr(t) associated with the random variable T is defined by hr(t) = lims-o- P(t ≤ T <t+8|T ≥ t) 8 Thus, we can interpret hr(t) as the rate of change of the probability that the object survives a little past time t, given that the object survives to time t. Show that if T is a continuous random variable, then hr(t) - fr(t) d 1 - Fr(t) dt —— log (1 - Fr(t)).
Transcribed Image Text:Suppose the random variable T is the length of life of an object (possibly the lifetime of an electrical component or of a subject given a particular treatment). The hazard function hr(t) associated with the random variable T is defined by hr(t) = lims-o- P(t ≤ T <t+8|T ≥ t) 8 Thus, we can interpret hr(t) as the rate of change of the probability that the object survives a little past time t, given that the object survives to time t. Show that if T is a continuous random variable, then hr(t) - fr(t) d 1 - Fr(t) dt —— log (1 - Fr(t)).
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