Final answer:
To show that 1/c equals the area of region R for a uniform distribution of a continuous random variable, one must integrate the joint density function over R, and the result should be 1. The constant c is then found to be 1/Area of R. For a square centered at (0,0) with sides of length 2, the constant c is 1/4 since the area of the square is 4.
Step-by-step explanation:
Uniform Distribution in Continuous Random Variables
To solve the mathematical problem completely, we must first understand the uniform distribution related to continuous random variables (RVs). By definition, a continuous random variable, such as (X, Y), uniformly distributed over a region R, implies that the probability of (X, Y) falling within any sub-region of R is proportional to the area of that sub-region. The joint density function f(x, y) for (X, Y) is given by:
f(x, y) = c if (x, y) ∈ R, 0 else
To show that 1/c = area of region R, we integrate the joint density function over the entire region R, which should equal 1 since the total probability must sum up to 1.
The f(x, y) integrated over region R equals:
∫∫_R f(x, y) dxdy = ∫∫_R c dxdy = c × (Area of R) = 1
Hence, we find that c = 1/Area of R. For a square centered at (0,0) with sides of length 2, the area of R is 2· 2 = 4, thus c = 1/4.
The probability density function is uniformly distributed and the graph of the pdf would resemble a rectangle with the height of the rectangle corresponding to the constant c.