(a) Give the characteristic function for the uniform distribution on the interval [?1, 1]. Here you need to compute the integral

p (t ) = -DO 36 where f(x) is the pdf of the uniform distribution on [-1, 1]. (b) The characteristic function of a certain random variable X is given by 4x (t) = = (2 cos(t) + 3 cos(2t) + j sin(2t)) . Compute the mean and variance of X through the characteristic func- tion. Note that in the section 4.8, Characteristic functions (Hsu 2014, p. 156), the complex number ? such that ? = -1 is denoted as j. A complex number is a number of the form a + bj, where a and b are real numbers and j is such that j? = -1. Recall that there is no real number r such that r' = -1. Then j is not a real number. Example: 2 + 3j is a complex number. The set of complex number is denoted C. Clearly, R c C. Generally the complex number j is denoted as i and then a + bj is also denoted as a + bi. Note also that for real numbers a, b, c, d, (a + bj) + ( c + dj) = (a+b) + (c+ d)j and (a + bj). (c + dj) = ac - bd + (ad + be)j