PDF, CDF and EDF

We usually use a regular hexagon to model a cell. Please see Fig. 1. The center of the cell locations at the origin O. The six vertices of the cell are Vn, n-0. 5. Vertices and V3 are on the x-axis. The cell radius is OV= R user equipments (UEs) are randomly and uniformly distributed in the cell, including the cell boundary. Let(Xn, be the coordinates of the nth UE for n -1, ..., N. We assume that X1, ..., Xv are i.i.d. random variables. Also, we assume that M, ..., Yv are i.i.d. random variables. (a) Find .XnXs(x,y). the joint PDF of Xn and n. (10 points) (b) Find Fx, (x), the cumulative distribution function (CDF) of X, (c) Find Fy, (y), the CDF of Yn. (10 points) (d) Write a computer program to plot the cell and mark the locations (e) Write a computer program to plot Fxn (r) and the empirical dis- (10 points) In the following three problems, set R 2 km and N 104 of all UEs. (10 points) tribution function (EDF) of (X1, ..., XN) in the same figure. (10 points 2 3 0 VA 4 V5 Figure 1: A cell (6) Write a computer program to plot Fy.() and the E DF of (y, , Yv) in the same figure. (10 points) We usually use a regular hexagon to model a cell. Please see Fig. 1. The center of the cell locations at the origin O. The six vertices of the cell are Vn, n-0. 5. Vertices and V3 are on the x-axis. The cell radius is OV= R user equipments (UEs) are randomly and uniformly distributed in the cell, including the cell boundary. Let(Xn, be the coordinates of the nth UE for n -1, ..., N. We assume that X1, ..., Xv are i.i.d. random variables. Also, we assume that M, ..., Yv are i.i.d. random variables. (a) Find .XnXs(x,y). the joint PDF of Xn and n. (10 points) (b) Find Fx, (x), the cumulative distribution function (CDF) of X, (c) Find Fy, (y), the CDF of Yn. (10 points) (d) Write a computer program to plot the cell and mark the locations (e) Write a computer program to plot Fxn (r) and the empirical dis- (10 points) In the following three problems, set R 2 km and N 104 of all UEs. (10 points) tribution function (EDF) of (X1, ..., XN) in the same figure. (10 points 2 3 0 VA 4 V5 Figure 1: A cell (6) Write a computer program to plot Fy.() and the E DF of (y, , Yv) in the same figure. (10 points)