Random Variable Probability Calculation

This problem requires finding the ratio $\frac{a}{b}$ for a random variable $X$ using its probability distribution and a condition involving its mean ($\mu$) and standard deviation ($\sigma$). The condition is $\sigma^2 + \mu^2 = 2$. The variable $X$ takes values $0, 1, 2, 3$.

Step 1: Probability Sum Constraint

The total probability must sum to 1. This provides a relationship between $a$ and $b$. $ P(X=0) + P(X=1) + P(X=2) + P(X=3) = 1 $ $ \frac{2a+1}{30} + \frac{8a-1}{30} + \frac{4a+1}{30} + b = 1 $ Combine the fractions:

• $ \frac{(2a+1) + (8a-1) + (4a+1)}{30} + b = 1 $
• $ \frac{14a+1}{30} + b = 1 $

Multiply by 30:

• $ 14a + 1 + 30b = 30 $
• $ 14a + 30b = 29 $ (Equation 1)

Step 2: Calculate Mean $E[X]$

The mean $\mu$ (or $E[X]$) is calculated as the sum of each value multiplied by its probability.

• $ \mu = E[X] = \sum_{i=0}^{3} x_i P(X=x_i) $
• $ E[X] = \left(0 \times \frac{2a+1}{30}\right) + \left(1 \times \frac{8a-1}{30}\right) + \left(2 \times \frac{4a+1}{30}\right) + \left(3 \times b\right) $
• $ E[X] = 0 + \frac{8a-1}{30} + \frac{8a+2}{30} + 3b $
• $ E[X] = \frac{16a+1}{30} + 3b $

From Equation 1, we know $30b = 29 - 14a$, so $3b = \frac{29 - 14a}{10}$. Substitute this into the expression for $E[X]$:

• $ E[X] = \frac{16a+1}{30} + \frac{3(29 - 14a)}{30} $
• $ E[X] = \frac{16a+1 + 87 - 42a}{30} = \frac{88 - 26a}{30} $
• $ \mu = E[X] = \frac{44 - 13a}{15} $

Step 3: Calculate $E[X^2]$

The second moment, $E[X^2]$, is calculated similarly using squared values.

• $ E[X^2] = \sum_{i=0}^{3} x_i^2 P(X=x_i) $
• $ E[X^2] = \left(0^2 \times \frac{2a+1}{30}\right) + \left(1^2 \times \frac{8a-1}{30}\right) + \left(2^2 \times \frac{4a+1}{30}\right) + \left(3^2 \times b\right) $
• $ E[X^2] = 0 + \frac{8a-1}{30} + 4 \times \frac{4a+1}{30} + 9b $
• $ E[X^2] = \frac{8a-1 + 16a+4}{30} + 9b = \frac{24a+3}{30} + 9b $

Substitute $b = \frac{29-14a}{30}$ from Equation 1:

• $ E[X^2] = \frac{24a+3}{30} + 9 \left(\frac{29-14a}{30}\right) $
• $ E[X^2] = \frac{24a+3 + 261 - 126a}{30} = \frac{264 - 102a}{30} $
• $ E[X^2] = \frac{44 - 17a}{5} $

Step 4: Apply Condition $\sigma^2 + \mu^2 = 2$

The variance is defined as $\sigma^2 = Var(X) = E[X^2] - \mu^2$.

Substituting this definition into the given condition $\sigma^2 + \mu^2 = 2$ simplifies the equation:

• $ (E[X^2] - \mu^2) + \mu^2 = 2 $
• $ E[X^2] = 2 $

Step 5: Solve for Parameter $a$

Equate the calculated $E[X^2]$ to 2 and solve for $a$.

• $ \frac{44 - 17a}{5} = 2 $
• $ 44 - 17a = 10 $
• $ 17a = 44 - 10 = 34 $
• $ a = \frac{34}{17} = 2 $

Step 6: Solve for Parameter $b$

Substitute the value $a=2$ into Equation 1 ($14a + 30b = 29$).

• $ 14(2) + 30b = 29 $
• $ 28 + 30b = 29 $
• $ 30b = 1 $
• $ b = \frac{1}{30} $

Step 7: Calculate the Ratio $\frac{a}{b}$

Finally, compute the ratio $\frac{a}{b}$ using the found values $a=2$ and $b = \frac{1}{30}$.

• $ \frac{a}{b} = \frac{2}{1/30} $
• $ \frac{a}{b} = 2 \times 30 = 60 $

Final Answer Calculation

The calculation shows that $\frac{a}{b} = 60$. This value corresponds to Option C.