Finding the Variance of Y for Related Random Variables
This solution explains how to find the variance of a random variable \(Y\) when it is related to another random variable \(X\), which follows a Binomial distribution. We will use the properties of variance and the given relationship between \(X\) and \(Y\).
Understanding the Problem Setup
We are given two random variables, \(X\) and \(Y\).
- The relationship between \(X\) and \(Y\) is defined by the equation:
\(\frac{2X-3Y}{5X+4Y} = 4\)
- The random variable \(X\) follows a Binomial distribution with parameters \(n=10\) and \(p=\frac{1}{2}\). This is denoted as \(X \sim Binomial(n=10, p=\frac{1}{2})\).
- Our goal is to determine the variance of \(Y\), denoted as \(Var(Y)\).
Step 1: Express Y in Terms of X
First, we need to simplify the given relation to express \(Y\) as a function of \(X\). This will allow us to use the properties of variance later.
- Start with the given equation: \(\frac{2X-3Y}{5X+4Y} = 4\)
- Multiply both sides by the denominator \((5X+4Y)\): \(2X - 3Y = 4(5X + 4Y)\)
- Distribute the 4 on the right side: \(2X - 3Y = 20X + 16Y\)
- Now, rearrange the equation to group terms involving \(Y\) on one side and terms involving \(X\) on the other side. Add \(3Y\) to both sides: \(2X = 20X + 16Y + 3Y\) \(2X = 20X + 19Y\)
- Subtract \(20X\) from both sides: \(2X - 20X = 19Y\) \(-18X = 19Y\)
- Finally, solve for \(Y\) by dividing both sides by 19: \(Y = -\frac{18}{19}X\)
This equation shows that \(Y\) is a linear transformation of \(X\). Specifically, \(Y = aX\) where \(a = -\frac{18}{19}\).
Step 2: Calculate the Variance of X
Since \(X\) follows a Binomial distribution \(Binomial(n, p)\), its variance is given by the formula:
\(Var(X) = np(1-p)\)Given \(n=10\) and \(p=\frac{1}{2}\), we can substitute these values into the formula:
\(Var(X) = 10 \times \frac{1}{2} \times \left(1 - \frac{1}{2}\right)\) \(Var(X) = 10 \times \frac{1}{2} \times \frac{1}{2}\) \(Var(X) = 10 \times \frac{1}{4}\) \(Var(X) = \frac{10}{4}\) \(Var(X) = \frac{5}{2}\)So, the variance of \(X\) is \(\frac{5}{2}\).
Step 3: Calculate the Variance of Y
We found that \(Y = -\frac{18}{19}X\). We can use the property of variance that states \(Var(aX) = a^2 Var(X)\), where \(a\) is a constant.
In our case, \(a = -\frac{18}{19}\). Applying this property:
\(Var(Y) = Var\left(-\frac{18}{19}X\right)\) \(Var(Y) = \left(-\frac{18}{19}\right)^2 Var(X)\)First, calculate the square of the constant:
\(\left(-\frac{18}{19}\right)^2 = \frac{(-18)^2}{19^2} = \frac{324}{361}\)Now, substitute the value of \(Var(X)\) we calculated:
\(Var(Y) = \frac{324}{361} \times \frac{5}{2}\)Multiply the fractions:
\(Var(Y) = \frac{324 \times 5}{361 \times 2}\)Simplify the expression. We can divide 324 by 2:
\(Var(Y) = \frac{162 \times 5}{361}\) \(Var(Y) = \frac{810}{361}\)Thus, the variance of \(Y\) is \(\frac{810}{361}\).
Conclusion
By simplifying the relationship between the random variables \(X\) and \(Y\) and applying the properties of variance for a Binomial distribution, we found that the variance of \(Y\) is \(\frac{810}{361}\).
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