Question

example 2: assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. find the probability of the given range of pounds lost.
a) ( p(8.3 leq x leq 9) = )
b) ( p(6 < x < 7.2) = )
c) ( p(x < 10.5) = )

Explanation:

Step1: Determine the formula for uniform distribution probability

For a uniform distribution \(X\sim U(a,b)\), the probability density function is \(f(x)=\frac{1}{b - a}\) for \(a\leq x\leq b\), and the probability \(P(c\leq X\leq d)=\frac{d - c}{b - a}\)

Given \(a = 6\) and \(b=12\)

Step2: Solve for part A

For \(P(8.3\leq X\leq9)\)

$$
LATEXBLOCK0
$$

Step3: Solve for part B

For \(P(6\lt X\lt7.2)\)

$$
LATEXBLOCK1
$$

Step4: Solve for part C

For \(P(X\lt10.5)\)

$$
LATEXBLOCK2
$$

Answer:

A. \(P(8.3\leq X\leq9)=\frac{7}{60}\approx0.117\)

B. \(P(6\lt X\lt7.2)=0.2\)

C. \(P(X\lt10.5)=0.75\)