Final answer:
The question discusses a uniform distribution U(0,1) for variable Y and defines a linear transformation for another variable W. The transformed variable W has a new uniform distribution U(a, b), where a and b are the new bounds. The formulas for mean and standard deviation are applicable to understand such distributions.
Step-by-step explanation:
The question revolves around the concept of a uniform distribution, which is a type of continuous probability distribution where all outcomes within the specified range are equally likely. In this case, the variable Y is given to have a uniform distribution U(0,1), which implies that Y can take any value between 0 and 1, both inclusive, with equal likelihood.
Furthermore, the variable W is defined as a linear transformation of Y, W=a+(b−a)Y, where a is the minimum value and b is the maximum value that W can obtain. Essentially, this transformation scales and translates the original uniform distribution of Y into another uniform distribution for W with the new bounds a and b.
The mean (μ) and standard deviation (σ) of a uniform distribution U(a, b) can be calculated using the formulas μ = (a+b)/2 and σ = √((b-a)²/12), respectively. These properties are fundamental to understanding and working with uniform distributions in probability and statistics.