1. State the problem: Complete the probability table and compute the mean $\mu$, the variance $\sigma^2$, and the standard deviation $\sigma$ for the given discrete distribution. 2. Given: The values and probabilities are $X=5,6,7$ with $P(X)=\frac{1}{5}$ each and $X=8,9,10,11$ with $P(X)=\frac{1}{10}$ each. 3. Check probabilities sum to 1 as a rule for valid distributions: $3\cdot\frac{1}{5}+4\cdot\frac{1}{10}=\frac{3}{5}+\frac{4}{10}=\frac{6}{10}+\frac{4}{10}=1$. 4. Formula for the mean (expected value): $$\mu=\sum X\,P(X)$$ 5. Compute each term $X\cdot P(X)$ and sum them to get $\mu$. 6. Calculations for $X\cdot P(X)$: $5\cdot\frac{1}{5}=1$. 7. $6\cdot\frac{1}{5}=1.2$. 8. $7\cdot\frac{1}{5}=1.4$. 9. $8\cdot\frac{1}{10}=0.8$. 10. $9\cdot\frac{1}{10}=0.9$. 11. $10\cdot\frac{1}{10}=1.0$. 12. $11\cdot\frac{1}{10}=1.1$. 13. Sum the $X\cdot P(X)$ values: $$\sum X\,P(X)=1+1.2+1.4+0.8+0.9+1.0+1.1=7.4$$ 14. Therefore the mean is $\mu=7.4$. 15. Formula for the variance: $$\sigma^2=\sum (X-\mu)^2\,P(X)$$ 16. Compute each $X-\mu$ and its square then multiply by $P(X)$. 17. For $X=5$: $X-\mu=5-7.4=-2.4$. 18. $(X-\mu)^2=(-2.4)^2=5.76$. 19. Contribution to variance: $5.76\cdot\frac{1}{5}=1.152$. 20. For $X=6$: $6-7.4=-1.4$. 21. $(X-\mu)^2=1.96$. 22. Contribution: $1.96\cdot\frac{1}{5}=0.392$. 23. For $X=7$: $7-7.4=-0.4$. 24. $(X-\mu)^2=0.16$. 25. Contribution: $0.16\cdot\frac{1}{5}=0.032$. 26. For $X=8$: $8-7.4=0.6$. 27. $(X-\mu)^2=0.36$. 28. Contribution: $0.36\cdot\frac{1}{10}=0.036$. 29. For $X=9$: $9-7.4=1.6$. 30. $(X-\mu)^2=2.56$. 31. Contribution: $2.56\cdot\frac{1}{10}=0.256$. 32. For $X=10$: $10-7.4=2.6$. 33. $(X-\mu)^2=6.76$. 34. Contribution: $6.76\cdot\frac{1}{10}=0.676$. 35. For $X=11$: $11-7.4=3.6$. 36. $(X-\mu)^2=12.96$. 37. Contribution: $12.96\cdot\frac{1}{10}=1.296$. 38. Sum the variance contributions: $$\sum (X-\mu)^2\,P(X)=1.152+0.392+0.032+0.036+0.256+0.676+1.296=3.84$$ 39. Therefore the variance is $\sigma^2=3.84$. 40. Standard deviation is $\sigma=\sqrt{\sigma^2}=\sqrt{3.84}$. 41. Simplify the square root: $\sqrt{3.84}=\sqrt{\dfrac{384}{100}}=\dfrac{\sqrt{384}}{10}=\dfrac{8\sqrt{6}}{10}=0.8\sqrt{6}\approx1.9595917942$. 42. Final answers: $\sum X\,P(X)=7.4$. 43. $\mu=7.4$. 44. $\sum (X-\mu)^2\,P(X)=3.84$. 45. $\sigma^2=3.84$. 46. $\sigma=0.8\sqrt{6}\approx1.9596$.