Question

Let \(X\) denote the lifetime of an item. Suppose the item has reached the age of \(t\). Let \(X_{t}\) denote its remaining life and define $$ \bar{F}_{t}(a)=P\left\\{X_{t}>a\right\\} $$ In words, \(\bar{F}_{t}(a)\) is the probability that a \(t\) -year-old item survives an additional time \(a\). Show that (a) \(\bar{F}_{t}(a)=\bar{F}(t+a) / \bar{F}(t)\) where \(F\) is the distribution function of \(X\). (b) Another definition of IFR is to say that \(F\) is IFR if \(\bar{F}_{t}(a)\) decreases in \(t\), for all \(a\). Show that this definition is equivalent to the one given in the text when \(F\) has a density.

Short answer

The relationship between \(\bar{F}_{t}(a)\) and \(\bar{F}(t+a) / \bar{F}(t)\) is given by \(\bar{F}_{t}(a) = \frac{\bar{F}(t + a)}{\bar{F}(t)}\). A distribution function \(F\) is IFR if its corresponding \(\bar{F}_{t}(a)\) decreases with \(t\). This definition is equivalent to the definition given in terms of the density function as proven above.

Step-by-step solution

Let's recall that the survivor function \(\bar{F}(x)\) is defined as the complement of the distribution function \(F(x)\): \(\bar{F}(x) = 1 - F(x)\). Now, we want to express \(\bar{F}_{t}(a)\) in terms of the survivor function \(\bar{F}(x)\). We have: $$ \bar{F}_{t}(a) = P\left\\{X_{t} > a \right\\} = P\left\\{X > t + a \mid X > t\right\\}. $$ Using the conditional probability definition, we get: $$ \bar{F}_{t}(a) = \frac{P\left\\{X > t + a, X > t \right\\}}{P\left\\{X > t\right\\}}. $$ Since if \(X > t + a\), it's implicit that \(X > t\), our equation becomes: $$ \bar{F}_{t}(a) = \frac{P\left\\{X > t + a\right\\}}{P\left\\{X > t\right\\}} = \frac{\bar{F}(t + a)}{\bar{F}(t)}. $$ This represents result (a). #Step 2: Show IFR definition equivalence for distribution with density#

We are given a definition of IFR as \(\bar{F}_{t}(a)\) decreasing in \(t\) for all \(a\). We need to show that this definition is equivalent to the one given in the text when \(F\) has a density. From the previous step, we know that \(\bar{F}_{t}(a) = \frac{\bar{F}(t + a)}{\bar{F}(t)}\). For \(\bar{F}_{t}(a)\) to be decreasing in \(t\), its derivative must be negative. Therefore, we need to find the derivative of \(\bar{F}_{t}(a)\) with respect to \(t\): $$ \frac{d \bar{F}_{t}(a)}{d t} = \frac{d}{dt} \left(\frac{\bar{F}(t + a)}{\bar{F}(t)}\right). $$ Using the Quotient Rule, we get: $$ \frac{d \bar{F}_{t}(a)}{d t} = \frac{\bar{F}'(t)\bar{F}(t + a) - \bar{F}(t)\bar{F}'(t + a)}{[\bar{F}(t)]^{2}}. $$ Now, recall that the derivative of a survivor function \(\bar{F}'(x)\) is related to the density function \(f(x) = -\bar{F}'(x)\). Substituting in the relationship between the derivative of a survivor function and the density function, we get: $$ \frac{d \bar{F}_{t}(a)}{d t} = \frac{f(t)\bar{F}(t + a) - f(t + a)\bar{F}(t)}{[\bar{F}(t)]^{2}}. $$ For the given definition of IFR to be equivalent to the one in the text, we need to show that the negative sign of the above derivative is consistent with the definition of IFR in terms of the density function. The definition of IFR in terms of the density function is given by: $$ \frac{f(t + a)}{f(t)} \geq \frac{\bar{F}(t + a)}{\bar{F}(t)}. $$ Rearranging this inequality, we get: $$ f(t)\bar{F}(t + a) - f(t + a)\bar{F}(t) \leq 0. $$ Comparing this inequality with the derivative of \(\bar{F}_{t}(a)\), we can see that the decreasing \(\bar{F}_{t}(a)\) with respect to \(t\) is equivalent to the given definition of IFR in terms of a density function, thus proving the equivalence between the two definitions. This represents result (b).

Key concepts

Conditional Probability

When we talk about conditional probability, we are referring to the probability of an event occurring given that another event has already occurred. It's a fundamental concept in statistics that helps us update our beliefs about the likelihood of certain outcomes based on new information. For instance, consider an item with a lifetime denoted by the variable X. If we know the item has already lasted t years, the conditional probability lets us calculate the chance that it will continue to function for an additional time a.

Formally, the conditional probability of event A, given event B, is notated as P(A|B) and is computed using the formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
where P(A ∩ B) is the probability of both events A and B happening together, and P(B) is the probability of event B. In our item's lifetime context, this translates to understanding the remaining life of an item, based on how long it has already lasted.

Increasing Failure Rate (IFR)

The Increasing Failure Rate (IFR) is a concept used to describe the reliability of items over time. It’s particularly useful in survival analysis and reliability engineering. An item's failure rate is considered to be increasing if the probability of failure in the next instant increases with the passage of time. This might mean that as an item gets older, it is more likely to fail.

To get a bit more technical, a distribution F is said to have an IFR if for every additional time period a, the probability of surviving that period decreases as the item grows older. If we let \( \bar{F}_t(a) \) denote the probability that an item of age t will survive an additional time a, then for a distribution to be IFR, \( \bar{F}_t(a) \) must be a decreasing function of t for all a. Practically speaking, this means that knowing the survival curve of an item can inform us about its likelihood of failing in the future more accurately as it ages.

Probability Density Function

A probability density function (PDF), denoted as f(x), is a function that describes the likelihood of a continuous random variable taking on a particular value. The PDF is integral in describing various aspects of probability distributions, as it can help to determine properties such as mean, median, mode, and variance.

For a continuous random variable X, the probability density function f(x) has the characteristic that the area under the curve of f(x) over an interval (a, b) gives the probability that X falls within that interval. This is mathematically represented as:
\[ P(a < X < b) = \int_{a}^{b} f(x) dx \]
The relationship between the probability density function and the survivor function is particularly important in our context. The survivor function \( \bar{F}(x) \) tells us the probability that a random variable X exceeds a certain value x, and it is related to the PDF through the formula \( \bar{F}(x) = 1 - F(x) \), where F(x) is the cumulative distribution function (CDF). Moreover, the derivative of the survivor function with respect to x is the negative of the PDF:
\[ f(x) = - \frac{d\bar{F}(x)}{dx} \]
This relationship is crucial when studying the failure rate of an item over time, as it allows us to connect the concepts of survival and the likelihood of failure—integral points when looking at conditional probabilities and increasing failure rates.