In usual notations If u=5x+20 and u = 100 and variance of u =40 then what are the mean and variance of x ?
The correct answer and explanation is :
Given the equations for ( u ):
- ( u = 5x + 20 ) (Equation 1)
- ( u = 100 ) (Equation 2)
- Variance of ( u = 40 )
We are tasked with finding the mean and variance of ( x ).
Step 1: Find the value of ( x ) from ( u )
From Equation 2, we know that ( u = 100 ). Now substitute this into Equation 1:
[
100 = 5x + 20
]
Solving for ( x ):
[
100 – 20 = 5x \quad \Rightarrow \quad 80 = 5x \quad \Rightarrow \quad x = \frac{80}{5} = 16
]
Thus, the mean of ( x ), denoted as ( \mu_x ), is 16.
Step 2: Find the variance of ( x )
The variance of a variable is related to how it scales under a linear transformation. In this case, the transformation from ( x ) to ( u ) is given by ( u = 5x + 20 ), which is a linear transformation with slope 5. We know that variance scales by the square of the coefficient when we transform a variable linearly.
From the given information, the variance of ( u ) is 40. The variance of ( u ) is related to the variance of ( x ) by the equation:
[
\text{Var}(u) = 5^2 \times \text{Var}(x)
]
Substitute the given variance of ( u ):
[
40 = 25 \times \text{Var}(x)
]
Solving for ( \text{Var}(x) ):
[
\text{Var}(x) = \frac{40}{25} = 1.6
]
Final Answer:
- The mean of ( x ) is ( \mu_x = 16 ).
- The variance of ( x ) is ( \text{Var}(x) = 1.6 ).
Explanation:
The problem involves a linear relationship between ( x ) and ( u ), given by ( u = 5x + 20 ). The value of ( x ) is directly determined by substituting the given value of ( u ) into the equation. Since the variance of ( u ) is related to the variance of ( x ) by the square of the coefficient in the linear transformation, we use this relationship to find the variance of ( x ). The key steps involve solving for the value of ( x ) and then applying the property of variance under linear transformations to obtain the final result.