1. **State the problem:** We are given a discrete random variable $X$ with probabilities $P(X)$ and need to find the Mean ($\mu$), Variance ($\sigma^2$), and Standard Deviation ($\sigma$). 2. **Recall formulas:** - Mean: $$\mu = \sum X \cdot P(X)$$ - Variance: $$\sigma^2 = \sum (X - \mu)^2 \cdot P(X)$$ - Standard Deviation: $$\sigma = \sqrt{\sigma^2}$$ 3. **Calculate Mean:** Given $X = \{1,2,3,4,5,6\}$ and $P(X) = \{0.15, 0.50, 0.90, 0.60, 0.50, 0.30\}$ Calculate $X \cdot P(X)$: $$1 \times 0.15 = 0.15$$ $$2 \times 0.50 = 1.00$$ $$3 \times 0.90 = 2.70$$ $$4 \times 0.60 = 2.40$$ $$5 \times 0.50 = 2.50$$ $$6 \times 0.30 = 1.80$$ Sum these: $$\mu = 0.15 + 1.00 + 2.70 + 2.40 + 2.50 + 1.80 = 10.55$$ 4. **Calculate Variance:** Calculate $X - \mu$: $$1 - 10.55 = -9.55$$ $$2 - 10.55 = -8.55$$ $$3 - 10.55 = -7.55$$ $$4 - 10.55 = -6.55$$ $$5 - 10.55 = -5.55$$ $$6 - 10.55 = -4.55$$ Calculate $(X - \mu)^2$: $$(-9.55)^2 = 91.2025$$ $$(-8.55)^2 = 73.1025$$ $$(-7.55)^2 = 57.0025$$ $$(-6.55)^2 = 42.9025$$ $$(-5.55)^2 = 30.8025$$ $$(-4.55)^2 = 20.7025$$ Calculate $(X - \mu)^2 \cdot P(X)$: $$91.2025 \times 0.15 = 13.680375$$ $$73.1025 \times 0.50 = 36.55125$$ $$57.0025 \times 0.90 = 51.30225$$ $$42.9025 \times 0.60 = 25.7415$$ $$30.8025 \times 0.50 = 15.40125$$ $$20.7025 \times 0.30 = 6.21075$$ Sum these: $$\sigma^2 = 13.680375 + 36.55125 + 51.30225 + 25.7415 + 15.40125 + 6.21075 = 148.887375$$ 5. **Calculate Standard Deviation:** $$\sigma = \sqrt{148.887375} \approx 12.204$$ **Final answers:** - Mean ($\mu$) = 10.55 - Variance ($\sigma^2$) = 148.89 - Standard Deviation ($\sigma$) = 12.20 Note: The probabilities given do not sum to 1, which is unusual for a probability distribution, but calculations proceed as requested.