Question

1. State the conditions for which two events are said to be independent.
2. What are the conditions for a function to be a random variable?
3. Define sample space and classify the types of sample space.
4. Define Joint and Conditional Probability.
5. Define Equally likely events, Exhaustive events and Mutually exclusive events.
6. State the theorem of total probability.
7. Define Random variable with a Example.

Answer 1

Answer:

1. Independent Events:

Two events A and B are said to be independent if the fact that one event has occurred does not affect the probability that the other event will occur.

If whether or not one event occurs does affect the probability that the other event will occur, then the two events are said to be dependent.

2. In the development of the probability function for a discrete random variable, two conditions must be satisfied:

(1) f(x) must be nonnegative for each value of the random variable, and

(2) the sum of the probabilities for each value of the random variable must equal one.

3. In probability, sample space is a set of all possible outcomes of an experiment.

1. A sample space can be finite or infinite.
2. A sample space can be discrete or continuous.
3. A sample space can be countable or uncountable.

4. Joint probability is the probability of two events occurring simultaneously. Marginal probability is the probability of an event irrespective of the outcome of another variable.

Conditional probability is the probability of one event occurring in the presence of a second event.

5. Equally likely events are events that have the same theoretical probability (or likelihood) of occurring. Example. Each numeral on a die is equally likely to occur when the die is tossed. Sample space of throwing a die: { 1, 2, 3, 4, 5, 6 }

One or more events are said to be exhaustive if all the possible elementary events under the experiment are covered by the event(s) considered together. In other words, the events are said to be exhaustive when they are such that at least one of the events compulsorily occurs.

Mutually exclusive events are things that can't happen at the same time. For example, you can't run backwards and forwards at the same time. The events “running forward” and “running backwards” are mutually exclusive. Tossing a coin can also give you this type of event.

6. The total probability rule (also called the Law of Total Probability) breaks up probability calculations into distinct parts. It's used to find the probability of an event, A, when you don't know enough about A's probabilities to calculate it directly. ... The total probability rule is: P(A) = P(A∩B) + P(A∩Bc).

7. A typical example of a random variable is the outcome of a coin toss. Consider a probability distribution in which the outcomes of a random event are not equally likely to happen. If random variable, Y, is the number of heads we get from tossing two coins, then Y could be 0, 1, or 2.