Gaussian distribution

4. The figure has two spherical shells and three different possible gaussian surfaces. If both shells have a charge of 5nC what is the electric field at the innermost gaussian surface? 10 1 Shell O R Gaussian surfacetwo zero-mean real-valued Gaussian random variables X and Y are called jointly Gaussian if and only if their joint density is 1 fxy (x, y) = (2.44) 27 \\ det E exp - 7 (x, y)2-1 (x, y)T ) where (for zero-mean random vectors) the so-called covariance matrix E is E = E [(X, Y) (x, Y)] =( OXY (2.45) OXY (a) Show that if X and Y are zero-mean jointly Gaussian random variables, then X is a zero-mean Gaussian random variable, and so is Y. (b) Show that if X and Y are independent zero-mean Gaussian random variables, then X and Y are zero-mean jointly Gaussian random variables. (c) However, if X and Y are Gaussian random variables but not independent, then X and Y are not necessarily jointly Gaussian. Give an example where X and Y are Gaussian random variables, yet they are not jointly Gaussian. (d) Let X and Y be independent Gaussian random variables with zero mean and variance o'x and ov, respectively. Find the probability density function of Z = X +Y. Observe that no computation is required if we use the definition of jointly Gaussian random variables ,2) GAUSSIAN DISTRIBUTION. a) [5] Write down an expression for a Gaussian distribution, Define each variable. b) (5] Modify the above expression to describe a normalized Gaussian (or simply. "normal") distribution. c) (i) [4] Calculate the value of a normal distribution at the mean.4. Suppose a species has a constant birth rate but the death rate is pro- portional to the population. For clarity by birth rate and death rate we mean births/deaths per unit of population. (a) (5 points) Write down the differential equation modeling this system. Your answer should include two constants B, D relating to the birth and death rate respectively. (b) (5 points) Suppose that B - 4.5, D - 1. Find the critical points of the system and analyze their stability