A random variable X takes values 0, 1, 2, 3 with probabilities frac2a + 130, frac8a - 130, frac4a + 130$\frac{2a + 1}{30}, \frac{8a - 1}{30}, \frac{4a + 1}{30}$, b respectively, where a, b in mathbbR$a, b \in \mathbb{R}$. Let mu$\mu$ and sigma$\sigma$ respectively be the mean and standard deviation of X such that sigma^2 + mu^2 = 2$\sigma^2 + \mu^2 = 2$. Then fracab$\frac{a}{b}$ is equal to:

### Related Formula textSum of probabilities: sum P(X=x_i) = 1$$\text{Sum of probabilities: } \sum P(X=x_i) = 1$$ textVariance formula: sigma^2 = E(X^2) - mu^2 implies E(X^2) = sigma^2 + mu^2 = 2$$\text{Variance formula: } \sigma^2 = E(X^2) - \mu^2 \implies E(X^2) = \sigma^2 + \mu^2 = 2$$ E(X^2) = sum x_i^2 P(x_i)$$E(X^2) = \sum x_i^2 P(x_i)$$ ### Core Logic Given Probability Distribution: ### Step 1: Set up variance equation We know sigma^2 + mu^2 = sum x_i^2 P(x_i) = 2$\sigma^2 + \mu^2 = \sum x_i^2 P(x_i) = 2$. 0^2left(frac2a+130right) + 1^2left(frac8a-130right) + 2^2left(frac4a+130right) + 3^2(b) = 2$$0^2\left(\frac{2a+1}{30}\right) + 1^2\left(\frac{8a-1}{30}\right) + 2^2\left(\frac{4a+1}{30}\right) + 3^2(b) = 2$$ frac8a-130 + frac16a+430 + 9b = 2$$\frac{8a-1}{30} + \frac{16a+4}{30} + 9b = 2$$ frac24a+330 + 9b = 2$$\frac{24a+3}{30} + 9b = 2$$ 24a + 270b + 3 = 60 implies 24a + 270b = 57$$24a + 270b + 3 = 60 \implies 24a + 270b = 57$$ Dividing by 3: 8a + 90b = 19 quad dots (1)$$8a + 90b = 19 \quad \dots (1)$$ ### Step 2: Total Probability Equation Sum of all probabilities equals 1: frac2a+130 + frac8a-130 + frac4a+130 + b = 1$$\frac{2a+1}{30} + \frac{8a-1}{30} + \frac{4a+1}{30} + b = 1$$ frac14a+130 + b = 1$$\frac{14a+1}{30} + b = 1$$ 14a + 30b + 1 = 30 implies 14a + 30b = 29 quad dots (2)$$14a + 30b + 1 = 30 \implies 14a + 30b = 29 \quad \dots (2)$$ ### Step 3: Solve the Linear System From (2), multiply by 3: 42a + 90b = 87$42a + 90b = 87$. Subtract (1) from this new equation: (42a + 90b) - (8a + 90b) = 87 - 19$$(42a + 90b) - (8a + 90b) = 87 - 19$$ 34a = 68 implies a = 2$$34a = 68 \implies a = 2$$ Substitute a = 2$a = 2$ back into (1): 8(2) + 90b = 19 implies 16 + 90b = 19 implies 90b = 3 implies b = frac130$$8(2) + 90b = 19 \implies 16 + 90b = 19 \implies 90b = 3 \implies b = \frac{1}{30}$$ We need fracab$\frac{a}{b}$: fracab = frac21/30 = 60$$\frac{a}{b} = \frac{2}{1/30} = 60$$ ### Pattern Recognition Notice that sigma^2 + mu^2$\sigma^2 + \mu^2$ is simply the second moment E(X^2)$E(X^2)$. Avoid calculating mu$\mu$ independently. Create a simultaneous system using E(X^2)=c$E(X^2)=c$ and sum p=1$\sum p=1$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Probability