Answer to Question #249363 in Statistics and Probability for ayush

Question

A cereal manufacturer is aware that the weight of the product in the box varies slightly from box to box. In fact, considerable historical data has allowed the determination of the density function that describes the probability structure for the weight (in ounces). In fact, letting X be the random variable weight, in ounces, the density function can be described as f(x) ={2/5, 23.75 ≤ x ≤ 26.25,

{0, elsewhere

(a) Verify that this is a valid density function.

(b) Determine the probability that the weight is smaller than 24 ounces.

(c) The company desires that the weight exceeding 26 ounces is an extremely rare occurrence.

Answer 1

a)

For probability density function f(x):

$\int^{\infin}_{-\infin}f(x)dx=1$

We have:

$\frac{2}{5}\int^{26.25}_{23.75}dx=\frac{2(26.25-23.75)}{5}=2\cdot2.5/5=1$

So, f(x) is probability density function.

b)

$P(X<24)=\frac{2}{5}\int^{24}_{23.75}dx=\frac{2(24-23.75)}{5}=0.1$

c)

$P(X>26)=\frac{2}{5}\int^{26.25}_{26}dx=\frac{2(26.25-26)}{5}=0.1$