Chapter 5: Q 5.36 (page 214)

Suppose that the life distribution of an item has the hazard rate function $λ (t) = t^{3}, t > 0$ . What is the probability that
$(a)$ the item survives to age $2 ?$
$(b)$ the item’s lifetime is between $. 4$ and $1.4 ?$
$(c)$ a $1 -$ year-old item will survive to age $2 ?$

Short Answer

Expert verified

Therefore the,

$(a) P (X \geq 2) \approx 0.0183 (b) P (0.4 \leq X \leq 1.4) \approx 0.6109 (c) P (X \geq \frac{2}{X} \geq 1) \approx 0.0235$

Step by step solution

Given information:

The life distribution of an item has the hazard rate function $λ (t) = t^{3}, t > 0$ .

Let $X$ denote the life distribution. First, we compute the probability function of the life distribution.

$F (t) = 1 - e x p (- \int_{0}^{t} λ (u) d u) F (t) = 1 - e x p (- \frac{1}{4} t^{4})$

Part (a) step 2 Explanation:

It is to compute $P (X \geq 2)$

$P (X \geq 2) = 1 - P (X < 2) = 1 - F (2) = e^{- 4} \approx 0.0183$

Part (b) step 3 Explanation:

The probability is given by

$P (0.4 \leq X \leq 1.4) = P (X \leq 1.4) - P (X \leq 0.4) = F (1.4) - F (0.4) = (1 - e^{- 0.9604}) - (1 - e^{- 0.0064}) = e^{- 0.0064} - e^{- 0.9604} \approx 0.6109$

Part (c) step 4 Explanation:

The probability that a $1 -$ year-old item will survive to age $2$ is

$P (X \geq \frac{2}{X} \geq 1) = \frac{P (X \geq 2)}{P (X \geq 1)} = \frac{1 - F (2)}{1 - F (1)} = \frac{e^{- 4}}{e^{\frac{- 1}{4}}} = e^{\frac{- 15}{4}} \approx 0.0235$

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Suppose that the life distribution of an item has the hazard rate function $λ (t) = t^{3}, t > 0$ . What is the probability that
$(a)$ the item survives to age $2 ?$
$(b)$ the item’s lifetime is between $. 4$ and $1.4 ?$
$(c)$ a $1 -$ year-old item will survive to age $2 ?$

Short Answer

Step by step solution

Given information:

Part (a) step 2 Explanation:

Part (b) step 3 Explanation:

Part (c) step 4 Explanation:

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Suppose that the life distribution of an item has the hazard rate functionλ(t)=t3,t>0. What is the probability that(a)the item survives to age2?(b)the item’s lifetime is between.4and1.4?(c)a 1-year-old item will survive to age2?

Short Answer

Step by step solution

Given information:

Part (a) step 2 Explanation:

Part (b) step 3 Explanation:

Part (c) step 4 Explanation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Suppose that the life distribution of an item has the hazard rate function $λ (t) = t^{3}, t > 0$ . What is the probability that
$(a)$ the item survives to age $2 ?$
$(b)$ the item’s lifetime is between $. 4$ and $1.4 ?$
$(c)$ a $1 -$ year-old item will survive to age $2 ?$