Question

A continuous random variable \(X\) has a probability density function \(f(x)\); the corresponding cumulative probability function is \(F(x) .\) Show that the random variable \(Y=F(X)\) is uniformly distributed between 0 and 1 .

Short answer

The random variable Y = F(X) has its cdf F_Y(t)= t, which implies it is uniformly distributed on [0,1].

Step-by-step solution

Understand the Given Functions

Notice that we have two functions: the probability density function (pdf) of the random variable X, denoted as f(x), and the cumulative distribution function (cdf) of X, denoted as F(x). The cdf is the integral of the pdf.

Define the Random Variable Y

The exercise requires showing that the new random variable Y = F(X) is uniformly distributed between 0 and 1. Recall that a random variable Y is uniformly distributed on the interval [0,1] if its cdf is F_Y(y) = y for y in [0,1].

Find the Cumulative Distribution Function of Y

To find the cdf of Y, calculate the probability that Y is less than or equal to some value t in [0,1]. This can be written as P(Y ≤ t) = P(F(X) ≤ t).

Use the Inverse Relationship of the CDF

Since F(X) is a non-decreasing function, P(F(X) ≤ t) equals the probability that X is less than or equal to F^{-1}(t). Thus, P(Y ≤ t) = P(F(X) ≤ t) = P(X ≤ F^{-1}(t)).

Apply the Definition of Cumulative Probability Function

The cumulative distribution function F(x) is defined such that F(x) = P(X ≤ x). Therefore, P(X ≤ F^{-1}(t)) = t. This implies that F_Y(t) = t for t in [0,1].

Conclude the Proof

Since F_Y(t) = t for t in [0, 1], by definition, Y is uniformly distributed on [0,1].

Key concepts

Probability Density Function

A Probability Density Function (pdf) represents the likelihood of a continuous random variable taking on a specific value within its range. Unlike discrete variables, the probability at any single point for continuous variables is zero. Instead, the pdf helps to find the probability that the variable falls within a certain interval.

Key aspects of pdf:

• The pdf is a non-negative function.
• The area under the curve is equal to one, representing the total probability.
• To find the probability that the variable falls within an interval \[a, b\], you integrate the pdf over that interval: \[ P(a \leq X \leq b) = \int_{a}^{b} f(x) dx \].

Cumulative Distribution Function

The Cumulative Distribution Function (cdf) of a continuous random variable X, denoted as F(x), describes the probability that X will take a value less than or equal to x. Essentially, the cdf accumulates the probabilities up to x.

Properties of the cdf:

• It is a non-decreasing function.
• The value ranges between 0 and 1.
• It is calculated by integrating the pdf from negative infinity to x: \[ F(x) = \int_{-\infty}^{x} f(t) dt \].
• If x is extremely small, F(x) approaches 0, and if x is extremely large, F(x) approaches 1.

To use a cdf to find the probability that X falls within the interval \[a, b\], subtract the cdf values: \[ P(a \leq X \leq b) = F(b) - F(a) \].

Uniformly Distributed Random Variable

A uniformly distributed random variable has all intervals of the same length on the distribution's support with equal probability. For a variable Y uniformly distributed over \[0, 1\], every outcome between 0 and 1 is equally likely.

Important characteristics:

• The pdf of Y is uniform and denoted as \[ f_Y(y) = 1 \text{ for } 0 \leq y \leq 1 \].
• The corresponding cdf is \[ F_Y(y) = y \text{ for } 0 \leq y \leq 1 \].

In our original exercise, we transform X into Y using Y = F(X). By showing that the cdf of Y is \[ t \text{ for } 0 \leq t \leq 1 \], it is evident that Y is uniformly distributed between 0 and 1. Thus, Y follows the properties of a uniformly distributed random variable.