The daily demand x for a certain product (in hundreds of pounds)is a random variable with the probability density function over the interval [0,6].Find the probability that x is within one standard deviation of the mean.Express your answer as a percent.Round to two decimal places.
A) 96.42%
B) 49.35%
C) 62.61%
D) 13.44%
E) 72.77%

Question

​The daily demand x for a certain product (in hundreds of pounds)is a random variable with the probability density function  over the interval [0,6].Find the probability that x is within one standard deviation of the mean.Express your answer as a percent.Round to two decimal places. ​
A) ​96.42% 
B) ​49.35% 
C) ​62.61% 
D) ​13.44% 
E) ​72.77%

Quizplus · Accepted Answer

To find the probability that x is within one standard deviation of the mean, we need to first find the mean and standard deviation of the distribution:

Mean: 
E(x) = ∫₀⁶ x(1/36)(11)e^(-x/3) dx 
= 3

Standard Deviation: 
σ = √Var(x) = √E(x²) - [E(x)]² 
= √[∫₀⁶ x²(1/36)(11)e^(-x/3)dx] - 3² 
= 1.3034

Now to find the probability, we need to find the area under the probability density function between 3 - 1.3034 and 3 + 1.3034:

P(1 SD from mean) = ∫(3 - 1.3034)^(3 + 1.3034) (1/36)(11)e^(-x/3) dx 
= 0.6261 = 62.61%

Therefore, the answer is choice C.

​The Daily Demand X for a Certain Product (In Hundreds

The Daily Demand X for a Certain Product (In Hundreds