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The Probability Density Function (PDF) is a fundamental concept in understanding continuous random variables. It provides a way to describe the likelihood of a random variable taking a specific range of values. For any random variable, the graph of its PDF helps visualize how probability is distributed over various outcomes. This helps in identifying where the random variable is most likely to be found.

For the random variable X in the exercise, the PDF is defined as:

• A constant value of \( \frac{1}{3} \) on the interval \( -1 < x < 2 \).
• Zero elsewhere, indicating there is no probability density outside this range.

This means that the random variable X is uniformly distributed over the interval \( -1 < x < 2 \). The flat nature of this distribution implies each value within the interval has an equal chance of occurring. By knowing the PDF, we can calculate probabilities related to X by integrating over the desired interval, giving us the cumulative distribution function (CDF).

Transforming a random variable is a crucial tool in probability and statistics. It helps analyze a new variable that is a function of an existing random variable. Here, we examine the transformation from X to Y, where \( Y = X^2 \). This square transformation changes the way probabilities are distributed across the range of Y.

In the exercise, the transformation of X to Y required computing \( P(X^2 \leq y) \) over different ranges of y:

• For \( 0 \leq y < 1 \), since \( X^2 \leq y \) relates to the range \( -\sqrt{y} \leq X \leq \sqrt{y} \), we used the properties of X's CDF to find probabilities.
• For \( 1 \leq y < 4 \), the condition \( X^2 \leq y \) splits into separate ranges for X, namely \( -\sqrt{y} \leq X < -1 \) and \( 1 < X \leq \sqrt{y} \).

These intervals are critical because they define where the transformed variable exists based on the original variable's distribution. It allows us to understand the probability structure of Y based on the known properties of X.

Integrals play a vital role in probability and statistics, especially in calculating the cumulative distribution function (CDF) from a given PDF. A CDF provides the total probability that a random variable X is less than or equal to a certain value, integrating all the probabilities over a specified range.

To compute the CDF of X, integrate its PDF over the interval of interest:

• For example, integrating the PDF of \( X \) from -1 to \( x \) (where \( -1 < x < 2 \)), the resulting CDF is \( F(x) = \frac{1}{3}(x+1) \).
• The CDF values at certain points provide a clear probability distinction: \( F(x) = 0 \) for \( x \leq -1 \), \( F(x) = 1 \) for \( x \geq 2 \).

By differentiating the CDF of Y that we obtained, we can find the PDF of Y. This relationship showcases how integrals and derivatives are interconnected in probability theory, transforming expectations into clear probabilities.