\(f(x)\), a continuous probability function, is equal to \(\frac{1}{12}\), and the function is restricted to \(0 \leq x \leq 12\). What is \(P(0 < x < 12)\)?

12.

Find the probability that \(x\) falls in the shaded area.

\(f(x)\), a continuous probability function, is equal to \(\frac{1}{3}\) and the function is restricted to \(1 \leq x \leq 4\). Describe \(P(x>\frac{3}{2})\).

5.2 The Uniform Distribution

Use the following information to answer the next ten questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes.

1.5

2.4

3.6

2.6

1.6

2.4

2.0

3.5

2.5

1.8

2.4

2.5

3.5

4.0

2.6

1.6

2.2

1.8

3.8

2.5

1.5

2.8

1.8

4.5

1.9

1.9

3.1

1.6

Table \(\PageIndex{2}\)

The sample mean = 2.50 and the sample standard deviation = 0.8302.

The distribution can be written as \(X \sim U(1.5, 4.5)\).

Find the probability that a randomly selected home has more than 3,000 square feet given that you already know the house has more than 2,000 square feet.

Use the following information to answer the next eight exercises. A distribution is given as \(X \sim U(0, 12)\).

Draw the graph of the distribution for \(P(x > 9)\).

31.

Find \(P(x > 9)\).

Use the following information to answer the next eleven exercises. The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years.

Find the third quartile of ages of cars in the lot. This means you will have to find the value such that \(\frac{3}{4}\), or 75%, of the cars are at most (less than or equal to) that age.

Sketch the graph, and shade the area of interest.

Find the value \(k\) such that \(P(x < k) = 0.75\).

The third quartile is _______

5.3 The Exponential Distribution

Use the following information to answer the next ten exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: \(X \sim Exp(0.2)\)

Use the following information to answer the next 16 exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.