Answer to Question #137793 in Statistics and Probability for John Kafs

Question

Customers at TAB are charged for the amount of salad the take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces. Let
Χ = Salad plate filling weight.
i. Find the probability density function of Χ
ii. What is the probability that a customer will take between 12 and 15 ounces of salad? iii. What is the probability that a customer will take fewer than 5 ounces of salad.
iv. Find Ε ( Χ) and Var ( Χ)

Answer 1

solution

i) PDF of X

For a uniform distribution,

$$f(x)=\frac{1}{b-a}$$

For X between 5 and 15 ounces:

$$f(x) =\frac{1}{15-5} =\frac{1}{10}$$

Answer: $f(x)=\frac{1}{10}$

ii) Probability that $12<X<15$

$$f(12<X<15)=(15-12)*\frac{1}{10} = \frac{3}{10}$$

answer: $f(12<x<15)=\frac{3}{10}$

iii) probability that $X<5$

The domain of X is the range of values where the PDF is valid. The domain of $X=[5,15]$

The PDF is not valid for values of $X<5$

answer: $f(X<5) = 0$

iv) E(X) and Var(X)

$$E(X) = \frac{b+a}{2}$$

$$= \frac{15+5}{2}= 10$$

$$Var(X)= \frac{(b-a)^2}{12}$$

$$= \frac{(15-5)^2}{12}=\frac{100}{12}=8.3333$$

answer: $E(X) = 10$ and $Var(X)=8.3333$