Question

For the proof of Theorem 4.8.1, we assumed that the cdf was strictly increasing over its support. Consider a random variable \(X\) with cdf \(F(x)\) that is not strictly increasing. Define as the inverse of \(F(x)\) the function $$ F^{-1}(u)=\inf \\{x: F(x) \geq u\\}, \quad 0

Short answer

The random variable \( F^{-1}(U) \), where \( U \) is uniformly distributed between 0 and 1, has a cumulative distribution function \( F(x) \), as per the steps outlined above.

Step-by-step solution

Understand the Problem

We are to prove that the random variable \( F^{-1}(U) \), where \( U \) is uniformly distributed between 0 and 1, has a cumulative distribution function \( F(x) \). Basically, we want to show that applying the inverse CDF of \( F(x) \) to a uniform distribution results in \( F(x) \) being the CDF of the resultant random variable.

Define CDF of Random Variable \( F^{-1}(U) \)

Let's denote \( Z = F^{-1}(U) \). Then compute the CDF of \( Z \), \( G(z) = P(Z ≤ z) \). We need to compute this to show that it equals \( F(x) \).

Compute CDF of Random Variable \( F^{-1}(U) \)

We have \( G(z) = P(Z ≤ z) = P(F^{-1}(U) ≤ z) = P(U ≤ F(z)) \). Since \( U \) follows uniform distribution over (0,1), the \( P(U ≤ F(z)) = F(z) \). Thus, \( G(z) = F(z) \).

Statement of Conclusion

From step 3, we find that the cumulative distribution function of the random variable \( F^{-1}(U) \) equals \( F(x) \). Thus we've shown that \( Z = F^{-1}(U) \) has cumulative distribution function \( F(x) \).

Key concepts

Cumulative Distribution Function

The cumulative distribution function (CDF) is a fundamental concept in probability and statistics that describes the probability that a random variable will take a value less than or equal to a certain point.

First and foremost, to comprehend the CDF, it's crucial to recognize that it's a non-decreasing function which levels off to 1 as we move to the right along the x-axis. Its primary job is to encapsulate the distribution of probabilities in one neat formula or graph.

For a random variable (X), the CDF is represented as (F(x)), defined by the probability (P(X leq x)). This function is crucial for understanding the behavior of random variables, because it tells us the chance of the variable falling within a certain range.

In our exercise, we explored the inverse CDF, which, rather than taking a value of (X) and giving us the cumulative probability up to that point, it takes a probability (between 0 and 1) and provides us with the corresponding value of (X). The importance of the CDF extends to various applications like statistical inference and hypothesis testing.

Uniform Distribution

A uniform distribution, in the context of probability, refers to a type of probability distribution where all outcomes are equally likely to occur within the defined range.

When we speak of a continuous uniform distribution, we generally indicate a distribution over an interval, say (0,1). The key property here is that the probability is constant at any point in that interval—it's uniform.

For a uniform distribution from (0,1), any number between 0 and 1 is as likely as any other number. Its CDF is beautifully linear over the support, increasing from 0 to 1 as (x) ranges from 0 to 1, which is used in the exercise to derive results about the inverse CDF.

Random Variables

Random variables are the linchpins of probability theory. They are essentially variables whose values are determined by the outcomes of random phenomena.

There are two types of random variables: discrete, which can take a countable number of distinct outcomes, and continuous, which can take infinitely many values.

Our focus in these problems is usually on continuous random variables, since we're deriving and applying CDFs which require an interval of numbers. The random variable (Z) in the exercise is defined by applying the inverse CDF to a uniform distribution, representing a transformation from the uniform random variable (U) to a new variable with its own distribution.

Probability

Probability is a measure of the likelihood that a particular event will occur. It's a value between 0 and 1, with 0 indicating the event will never occur and 1 indicating certainty.

It all comes down to assigning numbers to events based on how likely they are, and these numbers possess properties that allow us to compute probabilities for complex events based on simpler ones.

The concept of probability is inherent in the definition of the CDF which gives us the probability that our random variable (X) is less than or equal to some value. In the given exercise, we dealt with a specific probability, namely the uniformly distributed (U), where every value between 0 and 1 had an equal chance of occurring. This becomes the basis for deriving the desired CDF of our new random variable.