Download Lecture Note, July 8, 2014 Chih-Hsin Hsueh I. Random Variable • A

Lecture Note, July 8, 2014
Chih-Hsin Hsueh
I. Random Variable
• A random variable is a mapping from the sample space S to the real line R. That is,
for each simple event in the sample space a real number is assigned. The value of
the random variable depends on the result of the experiment (i.e., which simple event
occurs)
• Types of random variables
– Discrete random variable: A random variable is called a discrete random variableif it takes only a finite or countable number of different values.
– Continuous random variable: A random variable is called a continuous random
variableif it takes infinitely many values corresponding to an interval on the real
line.
• Probability distribution of a random variable X: The probability distribution of a
random variable X lists the possible values of the variable together with their respective
probabilities.
II. Discrete Random Variable:
• Notations for discrete random variable.
• Properties of the probability distribution of a discrete random variable X.
– 0 ≤ p(x) ≤ 1
1
–
P
p(x) = 1
• Example:
• Expectation (Mean/Expected value) and Standard deviation of a discrete random
variable:
Let X be a discrete random variable with probability distribution p(x) . Then the mean
(expectation/expected value), variance and standard deviation of X are given by the
following : (all the summation is taken over all possible values of X)
µ = E(X) =
X
xp(x)
X
σ 2 = V ar(X) =
(x − µ)2 p(x)
√
σ = Sd(X) = σ 2
– Example: Suppose X = number of dots on the uppermost face of a fair die. Find
1. the probability distribution of X
2. mean, variance and standard deviation of X
3. the probability that X is greater than 4
2
4. the probability that X is 3 or less
III. Binomial Distribution
• Binomial Experiment: An experiment is known as binomial if it has the following properties:
1. There is a fixed number of trials, n.
2. Each trial results in either of the two outcomes.
3. The probability of success, p, is same for each trial.
4. All trails are independent.
Under this setting, we are interested in the number of successes(X) observed in n trials.
• Binomial Random Variable: The number of successes under binomial setting is known
as a binomial random variable, which can take the values 0, 1, 2,...,n.
• Binomial Distribution: The probability distribution of a binomial random variable is
known as a binomial distribution.
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