Question

A large number of $230\Omega$ resistors are obtained by combining $120\Omega$ resistors with a standard deviation of $4.0\Omega$ and $110\Omega$ resistors with a standard deviation of $3.0\Omega$. The standard deviation of the $230\Omega$ resistors thus formed will be

• A. $3.5\Omega$
• B. $5.0\Omega$
• C. $3.0\Omega$
• D. $12.0\Omega$

Answer 1

The correct option is B $5.0\Omega$

Standard deviation of sum of two random variable

$st.dev.(X+Y)=\sqrt{Var\left(X\right)+Var\left(Y\right)+2Cov(X.Y)}$

as X and Y in the given problem are independent, so their covariance will be zero

$\therefore Cov(X,Y)=0$

as $\text{variance}=(\text{standard deviation}{)}^2={\sigma }^2$

$\therefore {\sigma }_{x+y}=\sqrt{{\sigma }_1^2+{\sigma }_1^2}=\sqrt{3^2+4^2}=5$