x and y are uniformly distributed over the interval [0,1]. Find the probability that |x−y|, the distance between x and y, is less than 0.4

Question

x and y are uniformly distributed over the interval [0,1]. Find the probability that |x−y|, the distance between x and y, is less than 0.4

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2 weeks 2022-01-07T03:57:06+00:00 1 Answer 0 views 0

0.72

Step-by-step explanation:

Given:

– x and y are uniformly distributed over the interval [0,1].

– |x−y|, the distance between x and y, is less than 0.4

Find:

Find the probability when |x−y| < 0.4

Solution:

The constrained area is the portion of the unit square between the lines:

y=x−0.4 and y=x+0.4 .

– That’s the R2 interval:

⟨x,y⟩ ∈ [0;1] × [max{ 0 , x−0.4 } ;min{ 1 , x+0.4 }]

– This can be subdivided into:

(   [ 0 ; 0.4) x [ 0 ; x + 0.4     )

⟨x,y⟩∈ ( U [0.4;0.6) ×[x−0.4;x+0.4) )

(    U [0.6;1) ×[x−0.4; 1)         )

– The area enclosed is two equal units of triangles and one square. Hence, we calculate the areas:

Area of triangle = 0.5*B*H

Area of triangle = 0.5*0.8*0.8 = 0.32

Area of parallelogram = 0.4*0.2 = 0.08

– Hence probability is:

Total Area = 2*0.32 + 0.08 = 0.72