Question 3 (20 points) (a) Write down the density function for a mixture of three normal distributions with density functions o(x; /1, of), 6(x; /2, 0?) and o(x; (3, 03). Describe its characteristics, including a brief illustration. (b) Assume the weights in this normal mixture distribution are 71, 12, 73, where 73 =1 - 71 - 72. Compute the mean of this mixture distribution. (c) Compute the variance of this normal mixture distribution. (d) Is this normal mixture distribution the same as the distribution of a random variable Y = _TX,, where X, ~ N(u;, o?)? Justify your