1. (8pts) Determine whether or not the table is a valid probability distribution of a discrete random variable. Explain fully. (a) (b) (c) (d) x P (x) x 2 0 2 4 P (x) 0.3 0.5 0.2 0.1 x 0.5 0.25 0.25 P(x) 0.4 0.6 0.8 1.1 2.5 4.1 4.6 5.3 0.16 0.14 0.11 0.27 0.22 x 15 19 100 111 222 P (x) 0.3 0.1 0.1 0.4 0.1 1 Problem 2. (12pts) A discrete random variable X has the following probability distribution: x 77 78 79 80 81 P (x) 0.15 0.15 0.20 Compute each of the following quantities. (a) (1pt) P (80) (b) (1pt) P (X > 80). (c) (1pt) P (X 80). (d) (3pts) The mean of X. (e) (4pts) The variance 2 of X. 0.40 0.10 (f) (2pt) The standard deviation . 2 Problem 3. (10pts) In a quality control inspection at a factory producing light bulbs, it is claimed that 95% of their bulbs meet the required quality standards. To investigate this claim, you randomly select 15 light bulbs for inspection. Let X be the number of bulbs meeting the quality standards in your sample. Find the following: (a) (2pts)The probability that at most 12 bulbs in the sample meet the quality standards. (b) (2pts) The probability that at least 12 bulbs in the sample meet the quality standards. (c) (2pts)The probability that exactly 12 bulbs in the sample meet the quality standards. (d) (2pts) Find the mean (expected value) of X. (e) (2pts) Find the variance of X. 3