et X and Y be independent Erlang random variables with common parameter λ and of order m and n, respectively. Is the random variable X+Y Erlang? If yes, enter below its order in terms of m and n using standard notation. If not, enter 0.
m+n
use convolution.
Well, let's see. Adding two independent Erlang random variables does indeed result in another Erlang random variable. However, the order of the resulting Erlang distribution depends on the sum of the orders of the individual variables. So, if m and n are the respective orders of X and Y, then the order of X + Y would be m + n.
In terms of standard notation, we can say that the order of X + Y is (m + n).
So, the answer is: (m + n).
The sum of two independent Erlang random variables with the same parameter λ is not an Erlang random variable.
The sum of two independent Erlang random variables is a gamma random variable, where the order of the gamma distribution is the sum of the individual orders of the Erlang random variables.
Therefore, the order of the random variable X+Y would be m+n in terms of m and n using standard notation.
To determine if the random variable X+Y is Erlang, we need to check if it follows the properties of an Erlang distribution.
The Erlang distribution is defined as the sum of m exponential random variables with the same rate parameter λ. So, if X and Y are independent Erlang random variables with common parameter λ and orders m and n respectively, we can calculate the order of X+Y.
The order of the Erlang distribution is the sum of the orders of the constituent random variables. Therefore, the order of X+Y will be m + n.
So the answer is: m + n.