Random Variables Definition:
• A random variable is a function of an outcome, X = f(ω).
• That is, each possible value of X represents an event that is a subset of the sample space for the given experiment.
• The domain of a random variable is the sample space Ω. Its range can be the set of all real
• numbers R, or only the positive numbers (0,+∞), or the integers Z, or the interval (0, 1),
• etc., depending on what possible values the random variable can potentially take.
Classify the following random variables as discrete or continuous:
• X: the number of automobile accidents per year in Maharashtra
• Y : the length of time to play 18 holes of golf.
• M: the amount of milk produced yearly by a particular cow.
• N: the number of eggs laid each month by a hen.
• P: the number of building permits issued each month in a certain city.
• Q: the weight of grain produced per acre.
For example, the sample space giving a detailed description of each possible outcome when three electronic components are tested may be written
S = {NNN,NND,NDN,DNN,NDD,DND,DDN,DDD}, where N denotes nondetective and D denotes defective. Sometimes, for ease of calculations, it is often important to allocate a numerical description to the outcome. For example, each point in the above sample space can be assigned a numerical value of 0, 1, 2, or 3 reflecting number of defective components in the outcome. Thus all the outcomes in event E = {DDN,DND,NDD} will be assigned value 2.
Example:
Q) Two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. Lets a define a random variable (i.e. a function) Y: number of red balls. The possible outcomes and the values y of the random variable Y are as follows
Sample Space y
RR 2
RB 1
BR 1
BB 0
Solution :
Consider an experiment of tossing 3 fair coins and counting the number of heads. Let X be the number of heads. X has to be an integer between 0 and 3. Since assuming each value is an event, we can compute probabilities,
 |
Example: Consider the simple condition in which components are specified to be defective or not defective. We can define the random variable X as X = 1, if the component is defective, 0, if the component is not defective.
Expected Value of a Discrete Random
Variable
Expectation or expected value of a random variable X is its mean, the average value.
 |
Example revisited:
Consider an experiment of tossing 3 fair coins and counting the number of heads. Let X be the number of heads. X has to be an integer between 0 and 3.
Thus, the expected value of X equals
E[x] = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8)
=3/2
Q) The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given x 0 1 2 3 4 f(x) 0.41 0.37 0.16 0.05 0.01 Find the average number of imperfections per 10 meters of this fabric
--->
Q) A coin is biased such that a head is three times as likely to occur as a tail.
Find the expected number of tails when this coin is tossed twice
--->
 |
