Question

5 The length (in centimeters) of the leaf of a certain plant is a continuous random variable with probability density function defined by f(x) = for x in 14 [3, 5.8]. Find the mean of the distribution, the standard deviation of the distribution, and the probability that the random variable is between the mean and 1 standard deviation above the mean. The mean of the distribution is cm. (Round to one decimal place as needed.)

Answer 1

Solution The given pdf is free) = 5 14 for 3<x<5.8 and we have to find @ mean S.D. and P (42X341+) :: mean = f(x) = S FCN),kod 58 s 2 a dre S 14 3 5.8 5 = [ 1. [ 15.8924 14 E (x) e - 4.4 .. mean = 4.4 To find variance we first find 5.8 3 5.8 2 E(x2) = 5.ge dae 3 3 प 14 3 (5.8) 3 C 3 3 ErX) 20.0133 Now vex 2 = E(X2)-[E (X)]? - 20.0133 -(4-4)2 - vex) 0.6533 the deviation is nothing but Now standard square root of variance. .'. S. DIX)= - 0.6533 0.808269. CS Scanned with CamScanner

S. DIX) = 0.8083 * Now plusxalto) = P(4.4 <x<5.2083) 5.2.083 S.2083 5 Ta s [x dae = 4.4 14 4.4 11 S [ 5.2083 -6,4] pluLX Sluto) = 0.288678 .: probo that random variable is bet the mean and I standard deviation above the mean is 0.288678 CS Scanned with CamScanner