Answer:
a) 
b) 
Explanation:
Previous concepts
The cumulative distribution function (CDF) F(x),"describes the probability that a random variableX with a given probability distribution will be found at a value less than or equal to x".
The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution".
Part a
Let X the random variable of interest. We know on this case that 
And we know the probability denisty function for x given by:
In order to find the cdf we need to do the following integral:
Part b
Assuming that 
, then the density function is given by:
And for this case we want this probability:
And evaluating the integral we got:
We are given a CSP with only binary can concentrate assume we run backtrackingSearch with ARC
Answer:
15.24°C
Explanation:
The quality of any heat pump pumping heat from cold to hot place is determined by its coefficient of performance (COP) defined as
Where Q_{in} is heat delivered into the hot place, in this case, the house, and W is the work used to pump heat
You can think of this quantity as similar to heat engine's efficiency
In our case, the COP of our heater is
Where T_{house} = 24°C and T_{out} is temperature outside
To achieve maximum heating, we will have to use the most efficient heat pump, and, according to the second law of thermodynamics, nothing is more efficient that Carnot Heat Pump
Which has COP of:
So we equate the COP of our heater with COP of Carnot heater
Rearrange the equation
Solve this simple quadratic equation, and you should get that the lowest outdoor temperature that could still allow heat to be pumped into your house would be
15.24°C
Explanation:
Ohm's law is used here. V = IR, and variations. The voltage across all elements is the same in this parallel circuit. (V1 =V2 =V3)
The total supply current is the sum of the currents in each of the branches. (It = I1 +I2 +I3)
Rt = (8 V)/(8 A) = 1 Ω . . . . supply voltage divided by supply current
I3 = 8A -3A -4A = 1 A . . . . supply current not flowing through other branches
R1 = (8 V)/(3 A) = 8/3 Ω
R2 = (8 V)/(4 A) = 2 Ω
R3 = (8 V)/(I3) = (8 V)/(1 A) = 8 Ω
V1 = V2 = V3 = 8 V
