Let $X = \{ 1,2,3 , \ldots , n , \ldots \}$

question 232

Essay

Let $X = \{ 1,2,3 , \ldots , n , \ldots \}$ be a discrete random variable with probability density function $f ( n ) = r ( 1 - r ) ^ { n - 1 }$ , where $0 < r < 1$ .(a) Show that $\sum _ { n = 1 } ^ { \infty } f ( n ) = 1$ . Explain the significance of the value 1.(b) The expected value of the random variable X is defined by $E ( X ) = \sum _ { n = 1 } ^ { \infty } n f ( n )$ . Show that $E ( X ) = \frac { 1 } { r }$ . The distribution of X is known as the geometric distribution.

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Definitions:

Product-oriented Layout

A layout where workstations and equipment are arranged based on the sequence of operations needed to produce a specific product efficiently.

Product-oriented Layout

A Product-oriented Layout organizes a production facility around the sequence of operations performed on a particular product line, optimizing efficiency for that product's manufacturing process.

Work Cells

Small, self-contained production units within a factory, designed to manufacture a specific product or component efficiently.

Direct Labour Cost

The cost associated with the work done by those directly involved in the manufacturing of products or the provision of services.