An express mail service charges a special rate for any package that weighs less

Question

An express mail service charges a special rate for any package that weighs less than 1 lb. Let x denote the weight of a randomly selected parcel that qualifies for this special rate. The probability distribution of x is specified by the density curve shown below. Use the fact that area of a trapezoid = (base)(average of two side lengths) to answer each of the following questions:

(a) What is the probability that a randomly selected package of this type weighs at most 0.6 lb? between 0.25 and 0.5 lb? at least 0.75lb?

P (0.25 x 0.5) =  
P (x 0.75) =

(b) It can be shown that x = 7/12 and 2x = 11/144. What is the probability that the value of x is more than 1 standard deviation from the mean value? (Round the answer to four decimal places.)
P (x is more than one standard deviation from the mean value) =

Explanation / Answer

cdf = integrate 0.5+x [0.5,x]

= 0.5x + x^2/2 [0.5,x]

= 0.5x + x^2/2 - 0.5^2 - 0.5^2/2

= 0.5x + x^2/2 - 0.125

b)

P(X < 0.6) = 0.5 * 0.6 + 0.6^2/2 -0.125

= 0.355

P(0.25 < x < 0.5) = P(X < 0.5 ) - P(X < 0.25)

= 0 -0

= 0

P(X >=0.75) = 1 - P(X < 0.75)

= 1 - 0.5 * 0.75 - 0.75^2/2 +0.125

= 0.21875

b)

value = 7/12 + 11/144 = 95/144

P(X > 95/144) = 1 - P(X <=95/144)

= 1 - 0.5 * 95/144 - (95/144)^2/2 +0.125

= 0.5775

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