Question

A machine produces 0, 1 or 2 defective items in a day with probabilities of \( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \) respectively. Then, the standard deviation of the number of defective items produced by the machine in a day is ...........

• A. $\frac{1}{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{4}$
• D. 1

Hint:

To calculate the standard deviation, first find the mean, then calculate the variance by using $E[X^2]$ and subtracting the square of the mean. Finally, take the square root of the variance.

Answer 1

The Correct Option is B

Let the random variable $X$ represent the number of defective items produced by the machine. The possible values of $X$ are 0, 1, and 2, with corresponding probabilities as given in the problem:
- $P(X = 0) = \frac{1}{4}$
- $P(X = 1) = \frac{1}{2}$
- $P(X = 2) = \frac{1}{4}$
We will first calculate the mean ($\mu$) and variance ($\sigma^2$), and then find the standard deviation ($\sigma$).
Step 1: Calculate the mean
The mean is given by: \[ \mu = E[X] = \sum x P(X = x) = 0 \times \frac{1}{4} + 1 \times \frac{1}{2} + 2 \times \frac{1}{4} = 0 + \frac{1}{2} + \frac{2}{4} = 1 \] Step 2: Calculate the variance
The variance is given by: \[ \sigma^2 = E[X^2] - (E[X])^2 \] First, calculate $E[X^2]$: \[ E[X^2] = 0^2 \times \frac{1}{4} + 1^2 \times \frac{1}{2} + 2^2 \times \frac{1}{4} = 0 + \frac{1}{2} + \frac{4}{4} = 1.5 \] Now, calculate the variance: \[ \sigma^2 = 1.5 - (1)^2 = 1.5 - 1 = 0.5 \] Step 3: Calculate the standard deviation The standard deviation is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} = \sqrt{0.5} = \frac{1}{\sqrt{2}} \] Thus, the standard deviation is $\frac{1}{\sqrt{2}}$.