Question

While conducting experiments, a marine biologist selects water depths from a uniformly distributed collection that vary between 2.00 m and 7.00 m. What is the probability that a randomly selected depth is between 2.25 m and 5.00 m?

Answer 1

Answer:

The probability that a randomly selected depth is between 2.25 m and 5.00 m is 0.55.

Step-by-step explanation:

Let the random variable X denote the water depths.

As the variable water depths is continuous variable, the random variable X follows a continuous Uniform distribution with parameters a = 2.00 m and b = 7.00 m.

The probability density function of X is:

$f_{X}(x)=\frac{1}{b-a};\ a<X<b$

Compute the probability that a randomly selected depth is between 2.25 m and 5.00 m as follows:

$P(2.25<X<5.00)=\int\limits^{5.00}_{2.25} {\frac{1}{7.00-2.00} \, dx$

                               $=\frac{1}{5.00}\int\limits^{5.00}_{2.25} {1} \, dx\\\\=0.20\times [x]^{5.00}_{2.25} \\\\=0.20\times (5.00-2.25)\\\\=0.55$

Thus, the probability that a randomly selected depth is between 2.25 m and 5.00 m is 0.55.