Question

I have a 6 sided die with the numbers 1, 2, 3, 3, 4, 5 on it. I also have a hat filled with numbers. When I pick a number out of the hat, I get a number between 1 and 10. The hat number has a mean of 2 and a standard deviation of 2.5. I make a new random variable by combining these two previous random variables. The new random variable is made by taking the number I get from the hat and multiplying it by 7 and then adding 3. Then I add the number I rolled on the die. For example if I pick a 6 out of the hat and roll a 2 on the die my new random variable is 7(6) + 3 + 2 = 47.

What is the mean and standard deviation of the new random variables?

Answer 1

The out comes of a 6 sided die are 1,2,3,3,4,5.

The probability of each outcome is equally likely.

let x be the random variable that shows the out come of 6 sided die.

The probability distribution of x is as follows:

x	1	2	3	3	4	5
P(x)	1/6	1/6	1/6	1/6	1/6	1/6

This can again written as follows:

x	1	2	3	4	5
p(x)	1/6	1/6	2/6	1/6	1/6

The mean of x:

  

  

Variance of x ,

Let Y be the number picked from the hat.

Mean E(Y)= 2 and   = 2.5

Then now random variable z.

Z= 7y+3+x

Mean of Z ,

E(Z)=E(7y+3+x)

=E(7y)+E(3)+E(x)

= 7E(y)+3+E(x)

=(7*2)+3+3

= 20

Var(Z)=Var(7y+3+x)

= 7² var(y)+var(3+x)

=49 var(y) +var(3)+var(x)

= 49 * (2.5)² + 0 + 5/3 (variance o constant is 0)

= 3695 / 12

standard deviation of Z = _z =

  _z =

  _z = 17.54755

The mean of new random variable = 20

standard deviation of new random variable = 17.5476