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Answer:
The probability that a fish caught by a fisherman is between 0.25 feet and 2 feet long is 0.4375.
Step-by-step explanation:
The probability density function or p.d.f. for the length x, in feet, of some type of fish caught by sport fishermen is :
[tex]p(x) = \left \{{{(\frac{1}{4})\ ;\ 0\leq x\leq 4 } \atop{0\ ;\ otherwise} \right.[/tex]
The probability that a fish caught by a fisherman is between 0.25 feet and 2 feet long is:
[tex]P(0.25\leq X\leq 2)=\int\limits^{0.25}_{2} {(\frac{1}{4})} \, dx \\=|(\frac{1}{4})x|^{2}_{0.25}\\=(\frac{1}{4})(2-0.25)\\=0.4375[/tex]
Thus, the probability that a fish caught by a fisherman is between 0.25 feet and 2 feet long is 0.4375.
Using the uniform probability distribution, it is found that there is a 0.4375 = 43.75% probability that a fish caught by a fisherman is between 0.25 and 2 feet long.
An uniform distribution has two bounds, a and b.
The probability of finding a value between c and d is:
[tex]P(c \leq X \leq d) = \frac{d - c}{b - a}[/tex]
The density function is:
[tex]p(x) = \frac{1}{4}, 0 \leq x \leq 4[/tex]
Which is an uniform distribution with [tex]a = 0,b = 4[/tex].
The desired probability is:
[tex]P(0.25 \leq X \leq 2) = \frac{2 - 0.25}{4 - 0} = 0.4375[/tex]
0.4375 = 43.75% probability that a fish caught by a fisherman is between 0.25 and 2 feet long.
A similar problem is given at https://brainly.com/question/17088600
