Asked Nov 10, 2019

Choose a number B at random from the interval [0, 1] with uniform density. Find the probability that

(a) 1/3 < B < 2/3
(b) |B − 1/2| ≤ 1/4
(c) B < 1/4 or 1 − B < 1/4
(d) 3B2 < B


Expert Answer

Step 1

For the provided information, the uniform distribution is B~U(a, b).

Here a=0 and b=1. The probability density function for uniform distribution is


a.  The required probability for 1/3<B<2/3 can be obtained as:


Image Transcriptionclose

1 -db b -а В - P 3 3 -db 1-0 2 1 3 3 1 3 0.333 len

Step 2

b. The required probability for |B-1/2|≤1/4 can be obtained as:


Image Transcriptionclose

1 В- 2 1 Р В- =P 4 1 1 =P 1 <B < - 2 4 2 4 3 P B< 4 4 db 1-0 3 1 4 4 =0.5 en

Step 3

c. The required probability for B < 1/4 or 1 &minus...


Image Transcriptionclose

PB 4 -db 1 1-8-)-1 В < 4 4 3 Then 1 3 +1 -P B< PB 4 4 =1 Thus, the required probability is 1


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