1. Mathematical expectation, or expected value, of a random variable is calculated by taking the sum of each possible outcome value multiplied by its probability of occurring. 2. For a game with multiple outcomes and their associated probabilities and payoffs, the expected value is the average amount a player can expect to win or lose over many plays. A positive expected value means the game favors the player, while a negative value means it is biased against the player. 3. Variance measures the average amount that values of a random variable deviate from the mean or expected value. Unfortunately, the document does not provide the formulas for variance or standard deviation of a random variable.

1. 1.5 Mathematical Expectation and Variance

2. If a random variable X takes on the values x1,x2,…,xk, with the probability f(x1), f(x2), …, f(xk), its mathematical expectation or expected value is  = x1· f(x1) + x2· f(x2) + … + xk· f(xk) Mathematical Expectation

3. Mathematical ExpectationSuppose a game has n outcomes A1, A2,…, An with corresponding probabilities p1, p2,…, pn where pi =P(Ai) and p1 + p2 + … + pn = 1. Suppose the pay off to the player on outcome Ai is ai, where a positive ai is a win for the player and a negative ai is a loss. Then expected value for the player will beE = a1p1 + a2p2 + ….+ anpnThis quantity is called the mathematical expectation for the player.

4. Mathematical ExpectationThe expectation E is the amount that the player can expect to win on the average each time the game is played.If E is positive then game is in the favor of the player.If E is negative the game is biased against the player.Suppose E = 0 then the game is said to be fair and a player’s winnings and losses should be about equal when the fair game is played a large number of times

5. Mathematical ExpectationDefinition: If the probabilities of obtaining the amounts a1, a2 ,…, or an are p1, p2, … and pn, then the mathematical expectation is E = a1p1 + a2p2 + …+ anpnNote:a’s are positive when they represent profit, winnings or gain and they are negative when they represent loss, deficits or penalties.

6. Mathematical ExpectationDefinition: The odds that an event will occur are given by the ratio of the probability that the event will occur to the probability that the event will not occur provided neither probability is zero. i.e. if p is the probability of the event A then odds that event will occurIf the odds for the occurrence of event A are a to b, where a and b are positive integers, then probability of event A is

7. Mathematical ExpectationThe average amount by which the values of the random variable deviate from mean is given byUnfortunately,

8. Variance of a Random Variable.

10. Standard deviation of a Random variable.