The amount of snowfall in feet in a remote region of Alaska in the month of January is a continuous random variable with probability density function Find the amount of snowfall one can expect in any given month of January in Alaska.
A)7.5 ft
B)4 ft
C)2.5 ft
D)3.5 ft

Question

The amount of snowfall in feet in a remote region of Alaska in the month of January is a continuous random variable with probability density function  Find the amount of snowfall one can expect in any given month of January in Alaska.
A)7.5 ft 
B)4 ft 
C)2.5 ft 
D)3.5 ft

Quizplus · Accepted Answer

To find the expected value of a continuous random variable, we integrate the product of the variable and its probability density function over its entire range. 
E[X] = ∫(x*f(x)) dx from 0 to infinity 
= ∫(x*e^(-x/4)/16) dx from 0 to infinity 
We can use integration by parts to evaluate this integral: 
u = x, du = dx 
dv = e^(-x/4)/16, v = -4e^(-x/4) 
∫(x*e^(-x/4)/16) dx = -xe^(-x/4) / 4 - ∫(e^(-x/4) / 4) dx 
= -xe^(-x/4) / 4 - 4e^(-x/4) / 4 + C 
= -(x + 16)e^(-x/4) / 4 + C 
Plugging in the limits of integration, we get: 
E[X] = [-(infinity + 16)e^(-infinity/4) - (0 + 16)e^(-0/4)] / 4 
Since e^(-infinity/4) = 0 and e^(-0/4) = 1, we get: 
E[X] = 4 
Therefore, the amount of snowfall one can expect in any given month of January in Alaska is 4 feet.

The Amount of Snowfall in Feet in a Remote Region