Final answer:
The density function of U = Y₁² + Y₂², where Y₁ and Y₂ are independent standard normal variables, follows a chi-square distribution with 2 degrees of freedom, also known as a Rayleigh distribution.
Step-by-step explanation:
The question involves finding the density function of the random variable U = Y₁² + Y₂², where Y₁ and Y₂ are independent standard normal random variables. This is a problem that involves the distribution of the sum of the squares of independent standard normal variables, which follows a chi-square distribution with two degrees of freedom, since each Yᵢ represents one degree of freedom. The density function of U is thus given by the formula for a chi-square distribution with 2 degrees of freedom:
f(u) = ¹⁄₎ e^{-¹⁄₂u}, u > 0
where ¹⁄₎ is the coefficient (1/2) raised to the power of the degrees of freedom divided by the factorial of one less than the degrees of freedom (in this case, 1/2), and e is the base of the natural logarithm. Because there are two variables being squared and summed, the degrees of freedom k is 2. Therefore, this is a special case of a chi-square distribution known as the Rayleigh distribution.