Question

Let X be a continuous random variable with pdf : f(x)={3x² 0 0 otherwise

​ Find: (a) E[X].__________________ (b) Var(X)______ c)E[3X-5X²+1]________ d)Var(E[3X-5X²+1]_____)

Answer 1

The question involves finding the expected value, variance, and related measures for a continuous random variable with a quadratic probability density function. The calculations involve integrating the function or its transformations weighted by the pdf.

Step-by-step explanation:

In this question, we deal with a continuous random variable X with a given probability density function (pdf), and we are tasked to find various statistical measures for this variable, such as the expected value, variance, and the expected value and variance of a given function of X.

1. E[X]: The expected value of the random variable X, which is calculated as the integral of x multiplied by its pdf over all possible values of x.
2. Var(X): The variance of the random variable X, which is the expected value of the squared deviation from the expected value, E[(X - E[X])^2], and can be found using the formula Var(X) = E[X^2] - (E[X])^2.
3. E[3X-5X²+1]: The expected value of the function of the random variable X, which is calculated by taking the integral of the function multiplied by the pdf of X over all possible values of X.
4. Var(E[3X-5X²+1]): There's a typo here, as it should be the variance of the function of X (not the variance of the expected value). We calculate this as Var(g(X)) = E[g(X)^2] - (E[g(X)])^2 where g(X) = 3X - 5X^2 + 1.

The distribution of X is not explicitly given but inferred from the pdf: it is a quadratic distribution bounded on an interval (which must be determined), since the pdf is a quadratic function.