Consider X be a continuous random variable with the probability density function

Question

Consider X be a continuous random variable with the probability density function f(x) shown below 0 1 Use this distribution to answer the questions below (give all answers to 3 decimal places): a) What is the value of f(1)? b) what is P(X = 1) ? Probability = c) What is P(XS 1)? Probability d) What is P(X s 0.5)? Probability = e) What is P(X 0.6)? Probability- 9) What is P(X 2 0.4)? Probability h) What is P(O.5 SX S 1.5) ? You can save a lot of effort by using the symmetry of the distribution. Probability i) What is P(0.4

Explanation / Answer

a) f(1) =1 (as area under curve =1 ; hence (1/2)*base*height =1 ; height =1*2/2 =1)

b)P(X=1)=0 (as for continuous distribution probability under a point is 0 cause area under a point is 0)

c)P(X<=1) =1/2 (as area under curve before 1 =1/2)

d)P(X<=0.5) =(1/2)*(0.5)*(0.5)=0.125

e) P(X<0.5)=0.125

f)P(X>0.6) =1-P(X<0.6)=1-((1/2)*(0.6)*(0.6))=1-0.18 =0.82

g)P(X>=0.4)=1-P(X<0.4)=1-((1/2)*(0.4)*(0.4))=0.92

h)P(0.5<X<1.5) =1-2*(P(X<0.5)=1-2*0.125 =0.750

i)P(0.4<X<1.7) =0.875

j) mode =1 (as maximum density is at 1)

k) median =1 (as half values are below 1 and half above)

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