15.2.7 One dimensional random variables and their characteristics

Problem: The one-dimensional random variable \( \xi \) is given by the distribution density \( P(x)=\gamma e^{a x^{2}+b x+c} \), where \( a=-3, b=-4, c=0, x_{1}=\frac{1}{3}, \quad x_{2}=\frac{4}{3} \). Find the constant \( \gamma \), the expected value, the distribution function and the probability that the value \( \xi \) belongs to the integral \( \left[x_{1}, x_{2}\right] \).