1. **Problem Statement:** Calculate the mean and variance of the discrete random variable $X$ given its values and frequencies. 2. **Data for the first distribution:** Values $X$: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 Frequencies $f$: 10, 5, 4, 3, 2, 1, 0, 0, 0, 0 3. **Formula for Mean ($\mu$):** $$\mu = \frac{\sum X_i f_i}{\sum f_i}$$ 4. **Formula for Variance ($\sigma^2$):** $$\sigma^2 = \frac{\sum f_i (X_i - \mu)^2}{\sum f_i}$$ 5. **Calculate total frequency:** $$N = 10 + 5 + 4 + 3 + 2 + 1 + 0 + 0 + 0 + 0 = 25$$ 6. **Calculate mean:** $$\sum X_i f_i = 6\times10 + 7\times5 + 8\times4 + 9\times3 + 10\times2 + 11\times1 = 60 + 35 + 32 + 27 + 20 + 11 = 185$$ $$\mu = \frac{185}{25} = 7.4$$ 7. **Calculate variance:** Calculate each term $(X_i - \mu)^2 f_i$: - $(6 - 7.4)^2 \times 10 = ( -1.4)^2 \times 10 = 1.96 \times 10 = 19.6$ - $(7 - 7.4)^2 \times 5 = ( -0.4)^2 \times 5 = 0.16 \times 5 = 0.8$ - $(8 - 7.4)^2 \times 4 = 0.6^2 \times 4 = 0.36 \times 4 = 1.44$ - $(9 - 7.4)^2 \times 3 = 1.6^2 \times 3 = 2.56 \times 3 = 7.68$ - $(10 - 7.4)^2 \times 2 = 2.6^2 \times 2 = 6.76 \times 2 = 13.52$ - $(11 - 7.4)^2 \times 1 = 3.6^2 \times 1 = 12.96$ Sum these: $$19.6 + 0.8 + 1.44 + 7.68 + 13.52 + 12.96 = 55.99999999999999 \approx 56$$ Variance: $$\sigma^2 = \frac{56}{25} = 2.24$$ 8. **Final answers:** - Mean $\mu = 7.4$ - Variance $\sigma^2 = 2.24$ This process illustrates how to compute the mean and variance for a discrete random variable using its values and frequencies.