Question

An certain brand of upright freezer is available in three different rated capacities: 16 ft³, 18 ft³, and 20 ft³. Let X = the rated capacity of a freezer of this brand sold at a certain store. Suppose that X has the following pmf.

$$ \begin{array}{l|ccc} x & 16 & 18 & 20 \\ \hline p(x) & 0.5 & 0.4 & 0.1 \end{array} $$

(a) Compute E(x), E(X²), and V(x).  

(b) If the price of a freezer having capacity X is 69X - 650, what is the expected price paid by the next customer to buy a freezer?

(c) What is the variance of the price paid by the next customer? 

(d) Suppose that although the rated capacity of a freezer is X, the actual capacity is h(X)=X-0.008x². what is the expected actual capacity of the freezer purchased by the next customer? 

Answer 1

Concepts and reason

A variable which can assume only countable number of real values and for which the value of the variable takes depends on chance is known as a discrete random variable.

If the random variable X is a one-dimensional discrete random variable taking at most a countable infinite number of values then its probabilistic behaviour at each real point is described by a function called the probability mass function.

It is denoted as,

Fundamentals

The probability mass function of a discrete random variable X is a function that should satisfy the following properties.

The mean and variance of the discrete random variable by using probability mass function is expressed as follows:

(a)

From the give information,

Let the random variable X represents the rated capacity of a freezer sold at a certain store.

Hence, the following table represent the probability distribution of the random variable X.

Hence, (mean), and (variance) are computed as follows:

(b)

From the given information, the price of a freezer having capacity X is.

The expected price paid by the next customer to buy a freezer is calculated as follows:

(c)

From the given information, the price of a freezer having capacity X is.

The variance of price paid by the next customer to buy a freezer is calculated as follows:

(d)

If the price of freezer having capacity X is, then the expected price by the next year customer to buy a freezer is,

Ans: Part a

Therefore, and.

Part b

Therefore,

Part c

Therefore,

Part d

Therefore,