Final answer:
For continuous random variables, probabilities are given by the area under the probability density function. Transformations of random variables involve calculating new density functions, often using change of variables methods. In the case of a uniform distribution, 'a' and 'b' represent the bounds of the distribution, and the density is uniform between these bounds.
Step-by-step explanation:
When dealing with continuous probability density functions (pdf), the probability of a random variable is defined as the area under the pdf graph. If X is a positive random variable with density f, the density of a transformation Y = aX + b, where a and b are constants, can be found by using change of variables in probability. For the squared random variable X², the density can be derived similarly, often requiring the method of transformation of variables.
To apply this concept to a square's side length when its area is uniformly distributed in [a,b], we would first find the distribution of the area and then transform it to find the distribution of the side length.
Continuous Probability Functions and Transformations
For a uniform distribution U(0, 12), 'a' represents the lower bound of the distribution while 'b' represents the upper bound. The probability density would be 1/(b-a) for 0 ≤ x ≤ 12, with the area indicating the probability. The probability that X is greater than a specific value, such as 9, would be calculated as P(X > 9) = (b - 9)/(b - a).
Remember that for any continuous random variable, the probability of a single point value, such as P(x = c) is always zero because a point has no area under a curve.