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Solution:-(1)Given data:Here arandom variable is modeled by exponential distribution =xbeta=theta=4and lambda=1//4Here, we have to findout the:-a) Complete the sentence:..distribution of ' x ':-Here, The distribution of ' x ' is ' 4 'because beta,theta are ' 4 ' so. The Answer is " 4 "with mu xx:-Here the distribution of x is same to mean.{:[:.mu x=" distribution "कx],[:.mu x=4]:}with sigma_(x):-Here the mu x value is equal to the sigma_(x) value.{:[:.mu x=sigma_(x).],[:.sigma_(x)=4]:}:. The distribution os x is 4 with mu x=4 and with sigma_(x)=4b) Now, we find:-P(1 <= x <= 5) value:-we know, beta=theta=4{:[=>lambda=1//4],[=>1//lambda=4=beta],[:.beta=lambda^(-1)=4]:}:. Aean is al80, beta=lambda^(-1)=4{:[:.P(1 <= x <= 5)=int_(1)^(5)f(x)*dx],[" we know, "f(x)=lambda*e^(-lambda(x))*dx". "],[=>p(1 <= x <= 5)=int_(1)^(5)1//4*e^(-(1)/(4)(x))*dx],[=>p(1 <= x <= 5)=int_( ... See the full answer