1. **State the problem:** We are given a discrete random variable $X$ with values and their probabilities $P(X)$. We 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:** Calculate $X \cdot P(X)$ for each $X$: - $1 \cdot 0.15 = 0.15$ - $2 \cdot 0.50 = 1.00$ - $3 \cdot 0.90 = 2.70$ - $4 \cdot 0.60 = 2.40$ - $5 \cdot 0.50 = 2.50$ - $6 \cdot 0.30 = 1.80$ Sum these values: $$\mu = 0.15 + 1.00 + 2.70 + 2.40 + 2.50 + 1.80 = 10.55$$ 4. **Calculate Variance:** First find $X - \mu$ for each $X$: - $1 - 10.55 = -9.55$ - $2 - 10.55 = -8.55$ - $3 - 10.55 = -7.55$ - $4 - 10.55 = -6.55$ - $5 - 10.55 = -5.55$ - $6 - 10.55 = -4.55$ Square each difference: - $(-9.55)^2 = 91.2025$ - $(-8.55)^2 = 73.1025$ - $(-7.55)^2 = 57.0025$ - $(-6.55)^2 = 42.9025$ - $(-5.55)^2 = 30.8025$ - $(-4.55)^2 = 20.7025$ Multiply each squared difference by $P(X)$: - $91.2025 \times 0.15 = 13.680375$ - $73.1025 \times 0.50 = 36.55125$ - $57.0025 \times 0.90 = 51.30225$ - $42.9025 \times 0.60 = 25.7415$ - $30.8025 \times 0.50 = 15.40125$ - $20.7025 \times 0.30 = 6.21075$ Sum these values: $$\sigma^2 = 13.680375 + 36.55125 + 51.30225 + 25.7415 + 15.40125 + 6.21075 = 148.887375$$ 5. **Calculate Standard Deviation:** $$\sigma = \sqrt{148.887375} \approx 12.20$$ **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 we followed the instructions as given.