The mean and the standard deviation are the same for which distribution?Question 4Answera.Uniform distributionb.exponential distribution

To determine for which distribution the mean and standard deviation are the same, we need to analyze the properties of the given distributions.

1. Uniform Distribution: For a uniform distribution defined over an interval $[a, b]$:

   • Mean $\mu = \frac{a + b}{2}$
   • Standard Deviation $\sigma = \frac{b - a}{\sqrt{12}}$

   Setting the mean equal to the standard deviation gives us: $\frac{a + b}{2} = \frac{b - a}{\sqrt{12}}$

2. Exponential Distribution: For an exponential distribution with rate parameter $\lambda$:

   • Mean $\mu = \frac{1}{\lambda}$
   • Standard Deviation $\sigma = \frac{1}{\lambda}$

   Here, it is clear that mean equals standard deviation since both equal $\frac{1}{\lambda}$.

Final Answer

The exponential distribution (option b) is the one where the mean and standard deviation are the same.