Question

(9 Marks) The kength of time required by students to complete a one-hour xam is a random variable with a probability density function (pdf) given by f(x) + OSX S1 a) Find the value and write down the new pilf (points) b) Find the probability that a randomly selected student will finish in less than half an hour. (3 points) c) Find the mean and standard deviation of the distribution (4 points) SD

Answer 1

sal f(x)= cn²ta ocasi . we know that integration af Pat over the range asx gives equal to so, da = 1 - cx? * * Ida- T-131-1 + 1 => c=105 Pdf /f(x) = 1.5x²+x so P Oos 0.5 0.5 3 + b probabiliny that randomly selected Student will finish exam in less than half (0.5) an haus x <0.5) = flada S(1-52? + x)da S (1.5 x² + x)da a)dn = TP(XC0.5) = 0.1875 1.5 0.5 0.1875

Mean = F(X) = af(a)da Elx) = jx. (1.58%+2) dn jø-5234 x?)da 1-5 P1+231 Elx) = E(x) = 0.708 D Standard deviation = Variance Variance = EX?)- [EC®)? Elx2) = x² f(x) dx = ( 22 (1.5x²+x) da EUR) = 5*5.-27. こ Variance = 0:55 F(X2) - [(x)7? using 0 & Variance = 0.55- (0.708)2 variance = 0.049 Standard duiation = Jo.049 Standard deviation = 0.22