Question

A local television station sells \(15-\mathrm{sec}, 30-\mathrm{sec}\), and 60 -sec advertising spots. Let \(x\) denote the length of a randomly selected commercial appearing on this station, and suppose that the probability distribution of \(x\) is given by the following table: $$ \begin{array}{lrrr} x & 15 & 30 & 60 \\ p(x) & .1 & .3 & .6 \end{array} $$ a. Find the average length for commercials appearing on this station. b. If a 15 -sec spot sells for $$\$ 500$$, a 30 -sec spot for $$\$ 800$$, and a 60 -sec spot for $$\$ 1000$$, find the average amount paid for commercials appearing on this station. (Hint: Consider a new variable, \(y=\) cost, and then find the probability distribution and mean value of \(y .\) )

Short answer

a) The average length for commercials on this station is 46.5 seconds. \nb) The average amount paid for commercials on this station is \$890.

Step-by-step solution

Find the average length for commercials

The average of a variable in probability is called the expectation or the expected value. It is calculated by multiplying each probable outcome with its probability and summing up these values. For this case, the expected length of the commercials, \(E(X)\), is calculated as follows: \[E(X) = x_1*p(x_1) + x_2*p(x_2)+ x_3*p(x_3) = 15*0.1 + 30*0.3 + 60*0.6\]

Calculate the values for \(E(X)\)

Simply calculate the individual elements of the equation from step 1 and add the results:\[E(X) = 1.5 + 9 + 36\]

Find the value of \(E(X)\)

Sum up the results from step 2 to find the total average length of the commercials:\[E(X) = 46.5 \, seconds\]

Define a new variable \(Y\) for cost

Now we treat the cost for each type of commercial as a new variable \(Y\), which is as follows:\[y = \begin{cases} 500, & \text{if}\ x=15 \ 800, & \text{if}\ x=30 \1000, & \text{if}\ x=60 \end{cases}\]

Calculate the expected commercial cost

Now we find the expectation of this new variable \(Y\), the same way as we calculated \(E(X)\):\[E(Y) = y_1*p(x_1) + y_2*p(x_2)+ y_3*p(x_3) = 500*0.1 + 800*0.3 + 1000*0.6\]

Find the value of \(E(Y)\)

Calculate the individual elements of the equation from step 5 and add the results:\[E(Y) = 50 + 240 + 600\]

Calculate \(E(Y)\)

Sum up the results from step 6 to find the total average cost of the commercials:\[E(Y) = \$890\]