If mean of the Poisson distribution is m, then standard deviation is equal to :

Q: If mean of the Poisson distribution is m, then standard deviation is equal to :

In a Poisson distribution, the mean (denoted as $ m $) and the variance are both equal to the parameter $ m $. This is a unique property of the Poisson distribution, where the mean and variance are identical. The standard deviation, which is a measure of the dispersion or spread of the distribution, is the square root of the variance. Therefore, the standard deviation of a Poisson distribution is given by: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{m} \] This directly supports Option 2 as the correct answer, which states that the standard deviation is $ \sqrt{m} $. Now, let's examine why the other options are incorrect: 1. **Option 1: $ m $** - This option suggests that the standard deviation is equal to the mean. However, in a Poisson distribution, while the mean and variance are equal, the standard deviation is the square root of the variance, not the mean itself. Therefore, this option is incorrect. 2. **Option 3: $ m^2 $** - This option implies that the standard deviation is the square of the mean. This is incorrect because the standard deviation is derived from the variance, which is $ m $, and the standard deviation is the square root of the variance, not the square of the mean. 3. **Option 4: $ m(m-1) $** - This option suggests a formula that does not relate to the standard deviation of a Poisson distribution. The expression $ m(m-1) $ does not have any basis in the properties of the Poisson distribution, where the variance is $ m $ and the standard deviation is $ \sqrt{m} $. Thus, this option is incorrect. In conclusion, the correct answer is Option 2, $ \sqrt{m} $, because it accurately reflects the relationship between the mean and the standard deviation in a Poisson distribution.

Question

Accepted Answer

In a Poisson distribution, the mean (denoted as $ m $) and the variance are both equal to the parameter $ m $. This is a unique property of the Poisson distribution, where the mean and variance are identical. The standard deviation, which is a measure of the dispersion or spread of the distribution, is the square root of the variance. Therefore, the standard deviation of a Poisson distribution is given by:

$$ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{m} $$

This directly supports Option 2 as the correct answer, which states that the standard deviation is $ \sqrt{m} $.

Now, let's examine why the other options are incorrect:

1. **Option 1: $ m $** - This option suggests that the standard deviation is equal to the mean. However, in a Poisson distribution, while the mean and variance are equal, the standard deviation is the square root of the variance, not the mean itself. Therefore, this option is incorrect.

2. **Option 3: $ m^2 $** - This option implies that the standard deviation is the square of the mean. This is incorrect because the standard deviation is derived from the variance, which is $ m $, and the standard deviation is the square root of the variance, not the square of the mean.

3. **Option 4: $ m(m-1) $** - This option suggests a formula that does not relate to the standard deviation of a Poisson distribution. The expression $ m(m-1) $ does not have any basis in the properties of the Poisson distribution, where the variance is $ m $ and the standard deviation is $ \sqrt{m} $. Thus, this option is incorrect.

In conclusion, the correct answer is Option 2, $ \sqrt{m} $, because it accurately reflects the relationship between the mean and the standard deviation in a Poisson distribution.