Question

consider a uniform distribution from a = 2 to b = 25. (a) find the probability that x lies between 6 and 11. (b) find the probability that x lies between 11 and 17. (c) find the probability that x lies between 8 and 15. (d) find the probability that x lies between 3 and 22. click the icon to see the definition of the uniform distribution. (a) the probability that x lies between 6 and 11 is (round to three decimal places as needed.)

Explanation:

Step1: Recall probability formula for uniform distribution

For a uniform distribution on the interval $[a,b]$, the probability density function is $f(x)=\frac{1}{b - a}$ for $a\leq x\leq b$, and the probability $P(c\leq X\leq d)=\frac{d - c}{b - a}$, where $a = 2$ and $b=25$.

Step2: Calculate probability for part (a)

For $c = 6$ and $d = 11$, we have $P(6\leq X\leq 11)=\frac{11 - 6}{25 - 2}=\frac{5}{23}\approx0.217$.

Step3: Calculate probability for part (b)

For $c = 11$ and $d = 17$, we have $P(11\leq X\leq 17)=\frac{17 - 11}{25 - 2}=\frac{6}{23}\approx0.261$.

Step4: Calculate probability for part (c)

For $c = 8$ and $d = 15$, we have $P(8\leq X\leq 15)=\frac{15 - 8}{25 - 2}=\frac{7}{23}\approx0.304$.

Step5: Calculate probability for part (d)

For $c = 3$ and $d = 22$, we have $P(3\leq X\leq 22)=\frac{22 - 3}{25 - 2}=\frac{19}{23}\approx0.826$.

Answer:

(a) $0.217$
(b) $0.261$
(c) $0.304$
(d) $0.826$