Understanding probability density functions (PDFs) is crucial for working with continuous random variables. A probability density function describes the likelihood of a continuous random variable taking on a given value within a range. For a continuous random variable X, the probability that X falls within a certain interval \([a, b]\) is found by integrating the PDF over that interval.

Probability Density Function Analysis

The problem provides the probability density function (PDF) for a random variable X as:

$$f\left( x \right) = \left\{ {\begin{array}{} {0.25}&{if\;1 \le x \le 5}\\ 0&{otherwise} \end{array}} \right.$$

• This function defines a uniform distribution over the interval \([1, 5]\). This means that within this specific interval, every value of X has an equal probability density of 0.25.
• Outside this interval (i.e., when \(x < 1\) or \(x > 5\)), the probability density is 0. This indicates that the random variable X cannot take values outside of the range \([1, 5]\).
• A quick check for a valid PDF: the total area under the curve must be 1. For this uniform distribution, the area is base \(\times\) height = \((5 - 1) \times 0.25 = 4 \times 0.25 = 1\), confirming it's a valid probability density function.

Probability Calculation for P(X ≤ 4)

To find the probability \(P(X \le 4)\), we need to integrate the probability density function \(f(x)\) from the lowest possible value of X up to 4. Since the random variable X is defined to have a non-zero probability density only for \(x \ge 1\), our integration will start from 1 and go up to 4.

Probability Calculation Steps

1. Identify the interval for calculation: We want to determine \(P(X \le 4)\). Given that the probability density function \(f(x)\) is non-zero only for \(x \ge 1\), the effective lower limit for our integration is 1, and the upper limit is 4. So, we integrate from 1 to 4.
2. Set up the integral: The probability \(P(X \le 4)\) is given by the definite integral of \(f(x)\) from the lower bound (1) to the desired upper limit (4):

   $$P(X \le 4) = \int_{1}^{4} f(x) \, dx$$

3. Substitute the function's value: In the interval \([1, 4]\), the probability density function \(f(x)\) is \(0.25\). Therefore, the integral becomes:

   $$P(X \le 4) = \int_{1}^{4} 0.25 \, dx$$

4. Evaluate the definite integral:

   $$P(X \le 4) = [0.25x]_{1}^{4}$$

   Now, substitute the upper and lower limits into the expression:

   $$P(X \le 4) = (0.25 \times 4) - (0.25 \times 1)$$

   $$P(X \le 4) = 1.00 - 0.25$$

   $$P(X \le 4) = 0.75$$

5. Convert the decimal to a fraction: The probability \(0.75\) can be expressed as a fraction:

   $$0.75 = \frac{75}{100} = \frac{3 \times 25}{4 \times 25} = \frac{3}{4}$$

Probability Result Summary

Thus, for the given random variable X and its probability density function, the probability \(P(X \le 4)\) is \(\frac{3}{4}\).

This problem illustrates a fundamental application of probability density functions for continuous random variables, especially for a uniform distribution where probabilities are found by calculating the area under the constant density curve within the specified range.