Answer :
Answer:
(a) The random variable X is a continuous random variable.
(b) The probability density function is shown below.
(c) The probability that a 275 gallon tank gets less than 272.5 gallons or more than 273.8 gallons of water is 0.285.
Step-by-step explanation:
Let the random variable X be denoted as the amount of water filled by the machine in a 275 gallon tank.
(a)
The random variable X is a continuous random variable.
A continuous random variables assumes infinite values. That is, they assume values in a fixed interval. For example, the distance covered by a car.
A discrete random variable assumes fixed definite values. They assume whole number values. For example, number of customers visiting a bank in an hour.
The amount of water in the tank can be any value between 0 to 275 gallon.
Hence, the random variable X is a continuous random variable.
(b)
The probability density function of the continuous random variable X is given as follows:
[tex]0.25;\ 272<X<273[/tex]
[tex]f_{X}(x) =0.20;\ 273<X<274[/tex]
[tex]0.55;\ 274<X<275[/tex]
(c)
Compute the value of P (272.5 < X < 273.8) as follows:
[tex]P(272.5<X<273.8)=\int\limits^{273}_{272.5} {f(x)} \, dx+\int\limits^{273.8}_{273} {f(x)} \, dx \\=0.25\times [273-272.5]+0.20\times [273.8-273]\\=0.125+0.16\\=0.285[/tex]
Thus, the probability that a 275 gallon tank gets less than 272.5 gallons or more than 273.8 gallons of water is 0.285.