Question

If $m$ and ${\sigma ^2}$ are the mean and variance of variable X , whose distribution is given by :

X	$0$	$1$	$2$	$3$
P(X)	$\dfrac{1}{3}$	$\dfrac{1}{2}$	$0$	$\dfrac{1}{6}$

Then ,
A.$m = {\sigma ^2} = 2$
B.$m = 1,{\sigma ^2} = 2$
C.$m = {\sigma ^2} = 1$
D.$m = 2,{\sigma ^2} = 1$

Answer 1

Hint : Mean and variance is a measure of central dispersion. Central dispersion tells us how the data that we are taking for observation are scattered and distributed. Mean is the average of a given set of numbers. The average of the squared difference from the mean is the variance

Complete step-by-step answer :
Mean: Mean is the average of a given set of numbers.
Variance: The average of the squared difference from the mean is the variance.
Here in this question we have
Mean $= \sum {XP(X)}$
= 0 \times \dfrac{1}{3} + 1 \times \dfrac{1}{2} + 2 \times 0 + 3 \times \dfrac{1}{6}$$$$ = 1
Therefore we get mean $m = 1$.
Variance ${\sigma ^2} = \sum {{X^2}P(X) - {m^2}}$
= 0 \times \dfrac{1}{3} + 1 \times \dfrac{1}{2} + 4 \times 0 + 9 \times \dfrac{1}{6} - 1$$$$ = 2
Therefore we get variance ${\sigma ^2} = 2$.
So, the correct answer is “Option B”.

Note : Some properties of the mean are given by:
1. If we increase individual units by k, then the mean will increase by k.
2. If we decrease individual units by k, then the mean will decrease by k.
3. If we multiply each unit by k, then the mean will be multiplied by k.
4. If we divide each unit by k, then the mean will be divided by k.
Properties of Variance
1.If the variance is zero, this means that each value of the set is equal to the mean value.
2.If the variance is small, it means that the observations are pretty close to the mean value and if the value is greater, the deviations of the observations are far from the mean value.
3.If each observation is increased by ‘$a$’ where $a \in \mathbb{R}$, then the variance will remain unchanged.
4.If each observation is multiplied by ‘$a$’ where $a \in \mathbb{R}$, then the variance will be multiplied by ${a^2}$ also.