{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Probability DISCRETE RANDOM VARIABLES

Probability DISCRETE RANDOM VARIABLES - DISCRETE RANDOM...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
DISCRETE RANDOM VARIABLES Achievement Standards:  90643 (3.3) part, external; 90646 (3.6) part, external Key words:  Sigma notation, Expected value, variance, winnings, profit, gain, return,                    independent, standard deviation. 1. SIGMA NOTATION This notation will be used at times during this topic. Examples: A= { 1,3,5,7,9,11,……)        Σ i = 1 5 a 1   =  1  +  3   +  5   + 7   + 9   =  2 5                              Σ i = 1 5 (4 i + 1) = 5 + 9 + 13 + 17 + 2 1 = 6 5     2. EXPECTATION For a given probability function the mean ( μ ) or expected value of X  (E[X]) is given by the formula:  E[X] =  Σ   x i p(x i ) Example 1: Spinner       A probability distribution for the result of spinning this spinner is: In this example the mean, or expected, value is:  E[X] = 0  ×  0.1 + 1  ×  0.3 + 2  ×  0.4 + 3  ×  0.2 = 1.7 Example 2: Raffle A raffle has 100 tickets and has a first prize of $200, second prize of $100 and a third prize of $50. Prizewinners are  drawn, without replacement, in order (1 st , 2 nd , 3 rd ). Let X be a random variable representing the winnings  of a single ticket buyer.  E[X] = 200  ×  0.01 + 100  ×  0.01 + 50  ×  0.01 + 0  ×  0.9   = $ 3.50 x 0 1 2 3 P(X=x) 0.1 0.3 0.4 0.2 x 200 100 50 0 P(X=x) 0.01 0.01 0.01 0.97 Nulake p 147 Sigma p121, Ex 7.01, Ex 7.02, 7.03 1 2 3 0
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
(Note that this means that if the raffle is to be fair tickets should sell at $3.50) Example 3: Car Insurance A racing car valued at $200 000 has the probability of being a total loss estimated at 0.002, a 50% loss with  probability 0.01, and a 25% loss with probability 0.1. What should the insurance company charge if it wants to make an average profit of  $1 000 per car that it insures? Let X be a random variable representing the amount that the company has to pay out. E[X] = $6 400, so the company would have to charge $7 400. 3. EXPECTED VALUE OF A LINEAR FUNCTION OF A RANDOM VARIABLE E[aX+b] = aE[X] + b Proof:  E[aX+b] =  Σ   p i (ax +b)                          =  Σ  ( p i  (ax i )    +  p i  ( b))                          =  Σ  ( p i  ax i )   Σ  (  p i   b)                          = a Σ  ( p i  x i )   + b Σ  (  p i ),    Σ  (  p i ) = 1,  Σ ax i  = a Σ x i                          = aE[X] + b Example:   The random variable X is defined by the following probability distribution:   giving E[X] = 1.7 Then the probability distribution for 3X+2 is: giving E[3X+2] = 7.1 Note that 3E[X]+2 = 3  ×  1.7 + 2                               = 7.1
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern