Question

4.60 The sum of two uniform random numbers. Generate two random numbers between 0 and 1 and take Y to be their sum. Then Y is a continuous random variable that can take any value between 0 and 2. The density curve of Y is the triangle shown in Figure 4.12. (a) Verify by geometry that the area under this curve is 1. (b) What is the probability that Y is less than 1? [Sketch the density curve, shade the area that represents the probability, then find that area. Do this for part (c) also.] (c) What is the probability that Y is greater than 1.5? (d) What is the probability that Y is greater than 0.5? Height = 1 0 2 FIGURE 4.12 The density curve for the sum Y of two random numbers, Exercise 4.60.

Answer 1

solution It is given that Y is a continuous random variable which is sum of two random variables that take value between o and I The density cwwe of Y is a triangle given below Height 1 2 O Part (as verify by geometry that area under this curve is = I x base x height Answer Area of triangle 2 In our case base = 2 height=t hina hence area = 1x2x1 = 1 2 1 x 2X=1 Hence proved Area under curve is

The density of can be mathematically also writen as fyL41- , y oc ys! En A 2-4 I< y < 2 and S 성 fylyl dy = free of curve Sydy + 5² (2-41 dy + Area : O + 1 = 1 2 b) fly. 1-2-4 reight = 1 f=lhis o 2 L area Probability Y is less than t = Shaded above Al (het)

A = I x C base of shaded triangle) x height 2 = 1 X 1 X 1 = 2 = IX IXI =1 2 2 Ans hence Ply<1] = 1 Mathematically Plyss] = { ty dy = [2 )! = 2 Using Ena c ) fly 1 = 2-4 height flyly 0.5 1 1.5 2 0 base = 0.5 Ply> 1-5] = Area of shaded region shaded = 1 (Base of shaded region ) x height of region = x 0.5 x 0.5 Jebe tentu Ans. 1 8 wir

Mathematically Ply>1.5}= S 2-y dy 1.5 = [2y - 42 = (4 - ) - ( 22 (3/2) = 2 - 3 + 1x 9 Ans 001LO -1-1 8 이 Pl4>0.5] = Area of shaded region below flyle 2-4 height = 1 fly) = 4 h=0.5 2 o 0.5 1 b=0.5 Area of shaded region = Total area of triangle pria unshaded triangle -1- 3x 0.5X0.5 = 7 8

hence ans Pły > 0.5] = 8 Mathematically PLy30-5 ] = Stydy + scazy, dy 095 1 2 + [] ay-y² 0.5 - ( 3 ) *(4-2)-(2+) 1-1 8 +/-/2 + 1 2. = 1-1= ? Ans 8 8