Chapter 5: Q. 5.17 (page 216)

If has a hazard rate function $λ_{X} (t)$ , compute the hazard rate function of $a X$ where $a$ is a positive constant.

Short Answer

Expert verified

The function's hazard rate is, $λ_{a X} (t) = \frac{1}{a} λ_{X} (t / a)$ .

Step by step solution

Determine the Random hazard variables.

The hazard rate of a random variable is,

$λ_{a X} (t) = \frac{f_{a X} (t)}{1 - F_{a X} (t)}$

We know that,

localid="1649618601408" $F_{a X} (t) = P (a X \leq t) = P (X \leq t / a) = F_{X} (t / a)$

Equation of the value.

So we get that,

$f_{a x} (t) = \frac{d}{d t} F_{a x} (t) = \frac{d}{d t} (F_{X} (t / a)) = \frac{1}{a} f_{X} (t / a)$

Finally, we have that,

$λ_{a X} (t) = \frac{f_{a X} (t)}{1 - F_{a X} (t)} = \frac{\frac{1}{a} f_{X} (t / a)}{1 - F_{X} (t / a)} = \frac{1}{a} λ_{X} (t / a)$

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If has a hazard rate function $λ_{X} (t)$ , compute the hazard rate function of $a X$ where $a$ is a positive constant.

Short Answer

Step by step solution

Determine the Random hazard variables.

Equation of the value.

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If has a hazard rate functionλX(t), compute the hazard rate function of aXwhere ais a positive constant.

Short Answer

Step by step solution

Determine the Random hazard variables.

Equation of the value.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

Mechanics Maths

Theoretical and Mathematical Physics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

If has a hazard rate function $λ_{X} (t)$ , compute the hazard rate function of $a X$ where $a$ is a positive constant.