Find the cumulative distribution function for amount of detergent delivered by t

Question

Find the cumulative distribution function for amount of detergent delivered by the machine.

Determine the probability that more than 6.50 mL of concentrated detergent is delivered by the machine for every 100 mL of water.

What is the amount of detergent delivered by the machine such that only 10% of all amounts are exceeded by this amount?

Determine the mean amount of detergent delivered by the machine.

A special detergent, in concentrated form, has been developed for use in the machine so that a specified average amount is delivered for every 100 mL of water. Suppose the amount of detergent delivered by the machine is given by the random variable,x, which is uniformly distributed between 6.35 mL and 6.85 mL.  

Explanation / Answer

Ans:

Uniformely distributed between (6.35,6.85)

pdf of x:

f(x)=1/(6.85-6.35)=1/0.5=2

1)Cumulative distribution function(CDF):

F(x)=(x-6.35)/(6.85-6.35)=(x-6.35)/0.5

2)P(X>6.5)=1-F(6.5)

=1-(6.5-6.35)/0.5

=1-(0.15)*2

=1-0.3=0.7

P(X>6.5)=0.7

3)P(X>x)=0.1

1-(x-6.35)/0.5=0.1

0.9=2*(x-6.35)

0.45=(x-6.35)

x=6.35+0.45

x=6.8

4)mean=(6.35+6.85)/2=6.6

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