Let the density function f (x) of the random variable X be a continuous function and its distribution function f (x), then 2F (x) f (x) is a probability density function

Let the density function f (x) of the random variable X be a continuous function and its distribution function f (x), then 2F (x) f (x) is a probability density function

Yes. The probability density function is required to be nonnegative and the integral is 1
Because f (x) is nonnegative and f (x) is nonnegative, 2f (x) f (x) is nonnegative
∫ 2F (x) f (x) DX = ∫ 2F (x) DF (x) = f (x) ^ 2. Because f (x) is 0 at negative infinity and 1 at positive infinity, the integral value of F (x) ^ 2 is also 1. So 2F (x) f (x) is a probability density function