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 $Var(X)=8.3333$