# According to a study by Dr. John McDougall of his live-in weight

According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between 6 and 15 pounds a month until they approach trim body weight. Let's suppose that the weight loss is uniformly distributed. We are interested in the weight loss of a randomly selected individual following the program for one month. Give the distribution of X. Enter an exact number as an integer, fraction, or decimal.$f\left(x\right){=}_{}$ where $\le X\le .\mu =\sigma =$. Find the probability that the individual lost more than 8 pounds in a month.Suppose it is known that the individual lost more than 9 pounds in a month. Find the probability that he lost less than 13 pounds in the month.

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Step 1 Given, According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between 6 and 15 pounds a month until they approach trim body weight. Let's suppose that the weight loss is uniformly distributed. Step 2 The random variable $X=$ Weight loss in pounds Given $a=6,b=15$
$X\sim U\left(a,b\right)⇒X\sim U\left(6,15\right)$
$f\left(x\right)=\frac{1}{b-a}$
$=\frac{1}{15-6}$
$=\frac{1}{9}$
$\mu =\frac{a+b}{2}$
$=\left(6+\frac{15}{2}\right)$
$=\frac{21}{2}$
$=10.5$
$\sigma =\frac{b-a}{\sqrt{12}}$
$=\frac{15-6}{\sqrt{12}}$
$=\frac{9}{\sqrt{12}}$
$=2.5980$ Step 3 The probability that the individual lost more than 8 pounds in a month: $P\left(x>8\right)=\frac{b-x}{b-a}$
$=\frac{15-8}{15-6}$
$=0.7778$ (rounded off to 4 decimals) Step 4 Suppose it is known that the individual lost more than 9 pounds in a month. The probability that he lost less than 13 pounds in the month: $P\left(x<13\mu dx>9\right)=P\frac{99\right)$

$=\frac{\frac{13-9}{15-6}}{\frac{15-9}{15-6}}$
$=\frac{\frac{4}{9}}{\frac{6}{9}}$
$=\frac{2}{3}$
$=0.6667$ (rounded off to 4 decimals)