Friday, January 06, 2006

Maximum Likelihood

Stolen from wikipedia :)

What is likelihood? ( from mathworld)
Likelihood is the hypothetical probability that an event that has already occurred would yield a specific outcome. The concept differs from that of a probability in that a probability refers to the occurrence of future events, while a likelihood refers to past events with known outcomes.

Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set.

What is probability distribution?
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied

What is Expected Value?
In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff ("value")

For example, an American roulette wheel has 38 equally possible outcomes. A bet placed on a single number pays 35-to-1 (this means that you are paid 35 times your bet and your bet is returned, so you get 36 times your bet). So the expected value of the profit resulting from a $1 bet on a single number is, considering all 38 possible outcomes:

\left( -$1 \times \frac{37}{38} \right) + \left( $35 \times \frac{1}{38} \right),

What is Random Variable?
A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. For example, a random variable can be used to describe the process of rolling a fair die and the possible outcomes { 1, 2, 3, 4, 5, 6 }. Another random variable might describe the possible outcomes of picking a random person and measuring his or her height.

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