Question

Let a point be selected from the sample space \(\mathcal{C}=\\{c: 0

Short answer

The cumulative distribution function (CDF) of \(X\) is \(F_{X}(x) = 0\) for \(x < 0\), \(F_{X}(x) = \frac{1}{10} \sqrt{x}\) for \(x\) in [0, 100), and \(F_{X}(x) = 1\) for \(x \geq 100\). The probability density function (PDF) of \(X\) is \(f_{X}(x) = 0\) for \(x \leq 0\) or \(x \geq 100\), and \(f_{X}(x) = \frac{1}{20 \sqrt{x}}\) for \(x\) in (0, 100).

Step-by-step solution

Determine the Range of \(X\)

The range of the random variable \(X = c^{2}\) is \(R_{X} = [0, 100)\), as \(c\) ranges from 0 to 10 (not inclusive).

Compute the CDF

The cumulative distribution function (CDF) of \(X\) is defined as \(F_{X}(x) = P(X \leq x)\). So for \(x\) in the range 0 to 100, \(F_{X}(x) = P(c^{2} \leq x) = P(c \leq \sqrt{x}) = \int_{0}^{\sqrt{x}} \frac{1}{10} dz = \frac{1}{10}\sqrt{x}.\) For \(x\) outside the range, \(F_{X}(x)\) is 0 for \(x < 0\) and 1 for \(x \geq 100\).

Compute the PDF

The probability density function (pdf) of \(X\) is the derivative of its CDF. So, \(f_{X}(x) = \frac{dF_{X}}{dx} = \frac{d}{dx} \frac{1}{10}\sqrt{x} = \frac{1}{20\sqrt{x}}\) for \(x\) in the range (0,100) (excluding the boundaries where the pdf is undefined), and 0 otherwise.

Key concepts

Probability Density Function

The probability density function (pdf) is a crucial concept in statistics, essential for understanding continuous random variables. Unlike discrete random variables, which compute probabilities as sums, continuous variables require an integral-based approach.

The pdf helps us determine the likelihood of a random variable falling within a specific range. It is defined such that the integral of the pdf over an interval gives the probability that the random variable will lie within that interval. Mathematically, for a continuous random variable \(X\), the probability that \(X\) takes on a value in an interval \([a, b]\) is given by:
\[ P(a \leq X \leq b) = \int_a^b f_X(x) \text{ d}x \]
where \(f_X(x)\) is the pdf of \(X\). The pdf is always non-negative, and its integral over the entire space is one, reflecting the fact that the probability of \(X\) taking on any value in its space is certain.

In our exercise, the pdf is derived by taking the derivative of the cumulative distribution function (CDF), which provides a connection between these two functions. The result \(f_X(x) = \frac{1}{20\sqrt{x}}\) for \(x\) within the interval (0,100) signifies the probability density at each point \(x\). Outside this interval, the pdf is zero since the random variable \(X\), in this case, does not take on values outside the range [0, 100).

Random Variable

A random variable is a mathematical function that assigns a real number to each outcome of a random experiment. There are two types of random variables: discrete and continuous. In our context, we focus on a continuous random variable since it can take an infinite number of possible values within a certain range.

The random variable \(X\), defined by \(X(c) = c^2\), assigns a number \(c^2\) to each point \(c\) in the sample space. The sample space in this problem is the interval (0,10), and hence \(X\) can take values from 0 to 100. Random variables allow us to talk about probabilities in terms of numbers rather than outcomes of an experiment, which can often be more complex or less intuitive.

What's important to note is that through the random variable \(X\), we translate the original probability set function over \(c\) to a function of \(x\). This translation step is a core part of finding the CDF and pdf of \(X\) as it allows us to utilize calculus to find probabilities over the range of \(X\).

Probability Set Function

When working with probability, the set function describes how to assign probabilities to specific subsets of a sample space. For a continuous sample space, such as in our exercise, the probability set function is often expressed as an integral.

The probability set function provided, \(P(C) = \int_{C} \frac{1}{10} dz \), is a simple function where the probability of an event occurring within a subset \(C\) of the sample space is proportional to the length of \(C\), given that the total length of the sample space is 10. This mirrors the concept of uniform distribution within the specified range of the sample space \(\mathcal{C}\).

Understanding the probability set function is key to computing the CDF. The CDF, as calculated in the solution, provides a way to find the total probability of a random variable being less than or equal to a certain value. It essentially accumulates the probabilities provided by the set function for all possible event subsets up to the given point. This cumulative approach reflects the name 'cumulative distribution function.'