Computing The Expected Value and The Variance

Question

I've done the half of this though question but I couldn't figure out the other half. Any help would be much appreciated!

Mike is a salesperson for a large retailer and he is covering long distances with his car every day.
The number of highway tolls Mike will need to go through on any day can be modelled as a random
variable X and it is assumed that it has the PMF function shown​ below, (I computed the mean and the
variance, mean ​E(X) = 4.91 and variance = 0.9419). At each​ toll, Mike will be ticketed a ​$7 bill​ and, at
the end of the​ day, he will get a single ​$3 ​discount, thanks to his​ frequent-user discount card for
highway tolls.

k         6      5       4         3        2

Px(k)    0.3    0.4     0.25      0.01     0.04

Let the total highway costs incurred by Mike on a single day be modelled by another discrete random
variable T. Compute the expected value and variance of T.

Answer 1

You said that $\mathbb{E}(X) =4.91$ and $Var(X)=0.9419$ (I didn't check that they were correct).

Since each toll costs 7, $X$ is the number of tolls, and he has a single discount of 3 at the end of the day, we can say that the total cost incurred by Mike in a day will be $$ T = 7X - 3 .$$

Then, we can compute the mean and variance of $T$ using the properties of mean and variance:

$$ \mathbb{E}(T) = \mathbb{E}(7X-3) = 7 \mathbb{E}(X) -3 = 31.37 $$ $$ Var(T) = Var(7X-3) = 7^2 Var(X) = 46.1531 $$

EDIT: I used the following properties:

1. $ \mathbb{E}(X+Y) = \mathbb{E}(X) + \mathbb{E}(Y) $, for any couple of random variable $X$ and $Y$.
2. $ \mathbb{E}(a X) = a \mathbb{E}(X)$, for any random variable $X$, for any real number $a \in \mathbb{R}$.
3. $ Var(X+Y) = Var(X) + Var(Y) $, for random variable $X$ and $Y$ which are independent.
4. $ Var(aX) = a²Var(X)$, for any random variable $X$, for any real number $a \in \mathbb{R}$.
5. $ Var(a) = 0$, for any constant random variable $a \in \mathbb{R}$.