Probability Models and Random Variables

You pay $5 to play. Draw a card from the deck. If you draw the ace of hearts, you will $100 For any other ace you will $10 For any other heart you will $5 Any other card is a loser

XP(X=x)X – random variable; the different values of our random event

P(X=x) – probability of each value of the random variable. Must sum to 1.

E(X) ( )x P x

Expected Value(E(X)) - The value we would expect to get if we played the game many, many times. Theoretical long-run average.

E(X) ( )x P x

Variance (Var(X)) – the variance of a random variable is the expected value of the squared deviations from the mean.

Standard Deviation – square root of variance

22 Var(X) ( ) ( )x E X P x

SD(X) Var(X)

Game cost $5 to roll the die once If you roll a 6 you win $10 If you roll a 5 you win $7 If you roll a 3,4 you win $5 If you roll a 1,2 you lose. Find the expected value and standard

deviation.

XP(X=x)

Linear Transformations: For a random variable X, a new random variable Y may be created by applying a linear transformation, Y = aX + b, where a and b are constants. The mean, variance, and standard deviation become.

E(Y) = aE(X) + b Var(Y) = a2Var(X) SD(Y) = |a|SD(X)

Y Xa b 2 2 2Y Xa

Y Xa

For the random variable XE(X) = 12SD(X) = 3Find the new mean and standard deviation

given:

3X X – 5

-2X + 8

A new random variable may be made by finding the sum or difference of random variables. Let X and Y be independent random variables.

E(X + Y) = E(X) + E(Y)E(X – Y) = E(X) – E(Y)

Var(X + Y) = Var(X) + Var(Y)Var(X – Y) = Var(X) + Var(Y)

Standard Deviation:

X Y X Y

X Y X Y

2 2 2X Y X Y 2 2 2X Y X Y

2 2X Y X Y

E(X) = 10 SD(X) = 2E(Y) = 5 SD(Y) = 6Find mean and standard deviation of each

new random variable.X + Y

X - Y

In a game of dice a single die is rolled: a 6 gives 40 points, a 4 or 5 gives 10 points, and no points for a 1, 2, or 3. Find the mean and standard deviation.

Suppose the points are doubled. What is the new mean and standard deviation?

Suppose the game is played twice. What is new mean and standard deviation?