Verify that the functions are probability density functions...

Verify that the functions are probability density functions for a continuous random variable $X$ over the given interval. Determine the specified probability. $f(x)=\left\{\begin{array}{ll}{\frac{2}{x^{3}}} & {x>1} \\ {0} & {x \leq 1}\end{array}\right.$ over $(-\infty, \infty), P(4 \leq X<9)$

Verify that the functions are probability density functions for a continuous random variable $X$ over the given interval. Determine the specified probability.
$f(x)=\left\{\begin{array}{ll}{\frac{2}{x^{3}}} & {x>1} \\ {0} & {x \leq 1}\end{array}\right.$ over $(-\infty, \infty), P(4 \leq X<9)$

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Key Concepts

Probability Density Function

A probability density function (PDF) for a continuous random variable characterizes the likelihood that the variable takes on a particular value. The key properties of a PDF are that it is nonnegative for all values and that the integral of the function over the entire set of possible outcomes (its support) equals 1, ensuring that all probabilities sum to 1.

Normalization Condition

The normalization condition requires that the integral of the probability density function over its entire support must equal 1. This condition is essential to validate that a given function properly represents a probability distribution, as it confirms that the total probability across all possible values is fully accounted for.

Integration for Continuous Probabilities

For continuous random variables, probabilities are calculated by integrating the probability density function over the interval of interest. This process involves setting up and evaluating a definite integral, thereby obtaining the probability that the random variable falls within the specified interval.

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