The problem states:

1. The variance of the random variable $X$ is given as $\text{Var}(X) = 3$.
2. The relationship between $Y$ and $X$ is $Y = -5X - 1$.

Solution:

The variance of a transformed random variable $Y$ is computed using the formula:

$\text{Var}(Y) = a^2 \cdot \text{Var}(X),$

where $Y = aX + b$, and:

• $a$ is the coefficient of $X$,
• $b$ is a constant (does not affect the variance).

From $Y = -5X - 1$, we identify $a = -5$ and $b = -1$. Substituting into the formula:

$\text{Var}(Y) = (-5)^2 \cdot \text{Var}(X) = 25 \cdot 3 = 75.$

Final Answer:

$\text{Var}(Y) = 75$

Would you like further clarification or a more detailed breakdown?

Related Questions:

1. How does adding a constant $b$ to a random variable affect its variance?
2. Can you explain the general formula for the variance of a linear transformation of random variables?
3. What is the significance of variance in probability and statistics?
4. How would the calculation change if $Y = 5X + 1$?
5. If $X$ has a standard deviation of $\sqrt{3}$, what is the standard deviation of $Y$?

Tip:

Always remember that the variance depends on the square of the scaling factor but is unaffected by added constants.