Tesla Battery Recharge Time.The electric­ vehicle manufacturing company Tesla esti­mates that a driver who commutes 50 miles per day in a Model S will require a nightly charge time of around 1 hour and 45 minutes (105 minutes) to recharge the vehicle’s battery (Tesla company website). Assume that the actual recharging time required is uniformly distributed between 90 and 120 minutes.
a. Give a mathematical expression for the probability density function of battery rechar­ging time for this scenario.
b. What is the probability that the recharge time will be less than 110 minutes?
c. What is the probability that the recharge time required is at least 100 minutes?
d. What is the probability that the recharge time required is between 95 and 110 minutes?

Answer:

a.) f(x) = [tex]\frac{1}{30}[/tex] where 90 < x < 120

b.) [tex]\frac{2}{3}[/tex]

c.)  [tex]\frac{2}{3}[/tex]

d.)  [tex]\frac{1}{2}[/tex]

Step-by-step explanation:

Let

X be a uniform random variable that denotes the actual charging time of battery.

Given that, the actual recharging time required is uniformly distributed between 90 and 120 minutes.

⇒X ≈ ∪ ( 90, 120 )

a.)

Probability density function , f (x) = [tex]\frac{1}{120 - 90} = \frac{1}{30}[/tex] where 90 < x < 120

b.)

P(x < 110) = [tex]\int\limits^{110}_{90} {\frac{1}{30} } \, dx[/tex]

               = [tex]\frac{1}{30}[x]\limits^{110}_{90} = \frac{1}{30} [ 110 - 90 ] = \frac{1}{30} [ 20] = \frac{2}{3}[/tex]

c.)

P(x > 100 ) = [tex]\int\limits^{120}_{100} {\frac{1}{30} } \, dx[/tex]

                 = [tex]\frac{1}{30}[x]\limits^{120}_{100} = \frac{1}{30} [ 120 - 100 ] = \frac{1}{30} [ 20] = \frac{2}{3}[/tex]

d.)

P(95 < x< 110)  = [tex]\int\limits^{110}_{95} {\frac{1}{30} } \, dx[/tex]

                       = [tex]\frac{1}{30}[x]\limits^{110}_{95} = \frac{1}{30} [ 110 - 95 ] = \frac{1}{30} [ 15] = \frac{1}{2}[/tex]

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