Let X be a continuous random variable, uniformly distributed on some interval, and let Y = aX + b. The random variable will be a continuous random variable with a uniform distribution if and only if (choose one of the following statements):
a != 0
a >0 and b = 0
a != 0
Well, let me put it in a way that might make you smile. Imagine X is like a random number generator. Now, if we multiply X by a and add b to it to get Y, it's like giving X a fancy makeover.
To answer your question with a touch of humor, the random variable Y will have a uniform distribution if and only if "a" and "b" are the perfect ingredients that magically transform X into a charming, smooth-talkin' stud who can sweep any probability off its feet. So, statement (a) "a = 0, b ≠ 0" just won't cut it, because that would be like trying to turn X into Y without changing a thing – basically, like expecting a banana to grow into a pineapple without any effort.
On the other hand, statement (b) "a ≠ 0, b = 0" is like dressing up X in a fancy suit, but forgetting to give it that extra touch of charm. Sure, X might look all formal and serious, but it won't have that magical uniform distribution.
Therefore, the correct statement that turns X into Y with a uniform distribution is (c) "a ≠ 0, b ≠ 0". It's like adding some secret sauce to X, making it irresistibly appealing to probability distributions everywhere.
To determine whether the random variable Y = aX + b has a uniform distribution, we need to consider the transformation applied to the original distribution of X.
Option 1: a < 0
If a < 0, this means that the transformation will reflect and flip the original distribution of X. As a result, the transformed variable Y will not have a uniform distribution.
Option 2: a > 0 and b ≠ 0
If a > 0 and b ≠ 0, the transformation will preserve the shape of the original distribution but shift it along the x-axis. Although the shape remains the same, the interval over which Y is uniformly distributed will change. Therefore, Y will not have a uniform distribution in this case.
Option 3: a > 0 and b = 0
If a > 0 and b = 0, the transformation will stretch or compress the original distribution of X, while maintaining the location of the distribution on the x-axis. In this case, the transformed variable Y will have a uniform distribution over the same interval as X.
Therefore, the correct statement is: "the random variable will be a continuous random variable with a uniform distribution if and only if a > 0 and b = 0."