The correct answer is $\frac{{16}}{3}$.
A random variable is uniformly distributed over the interval 2 to 10 means that the probability of the random variable taking on any value between 2 and 10 is equal. The variance of a random variable is a measure of how spread out its values are. The formula for the variance of a uniformly distributed random variable is:
$$\sigma^2 = \frac{1}{12}(b-a)^2$$
where $a$ is the minimum value of the random variable and $b$ is the maximum value. In this case, $a=2$ and $b=10$, so the variance is:
$$\sigma^2 = \frac{1}{12}(10-2)^2 = \frac{1}{12}(8)^2 = \frac{{16}}{3}$$
Option A is incorrect because it is the variance of a normally distributed random variable with mean 6. Option B is incorrect because it is the variance of a binomial random variable with $n=6$ and $p=0.5$. Option C is incorrect because it is the variance of a Poisson random variable with mean 4. Option D is incorrect because it is the variance of a geometric random variable with mean 3.