Chapter 4
A random variable is a variable that assumes values associated with the random outcomes of a random experiment, where one and only one numerical
values is assigned to each sample point.
The distance between you home and UCF is between 0 and 100 miles that is an interval, i.e., the distance between your home and UCF is a continuous
random variable
The number of heads in coin tossing experiment is a count, i.e., the number of heads in a coin tossing experiment is a discrete random variable
Usually, we can use the following four steps to complete a probability table.
Step 1: Find out the variable of interest.
Step 2: List all the sample points in the sample space.
Step 3: List all the possible values of this random variable.
Step 4: Assign the probabilities to all the possible values.
The probability distribution of a discrete random variable is a graph, a table, or a formula that specifies the probability associated with each possible value
the random can assume. The probability distribution should not include values that have zero probabilities. Thus, the probability of any value of a random
variable is among 0 and 1 and the sum of the probabilities of all possible values of a random variable is equal to one.
Population Mean – μ = Σ xp(x) = E(x)
Population Variance – σ2= E[(x-μ)2] = Σ(x-μ)2x p(x)μ)2] = Σ(x-μ)2] = Σ(x-μ)2x p(x)μ)2x p(x)
Population Standard Deviation – σ=
√
σ2
Binomial Random Variable – characteristics: First, they consist of n identical and independent trials. Second, there are only two possible outcomes denoted
by S and F on each trail. Third, the possibility of each outcome remains unchanged from trial to trial, that is, the probability of S is p and probability of F is
q=(1-μ)2] = Σ(x-μ)2x p(x)p). Fourth, we are interested in the random variable x represented the number of S happened in n trails (n is a fixed number). Therefore, it is worth to
develop a special probability model to deal with this kind of random variables. Any random variable that has these four characteristics is called binomial
random variable and can be dealt by using this special probability model.
Suppose that X is a binomial random variable. The probability of success on any single trial is p and there are n trials in this random experiment. The
probability density function of X is P(X=x) = p(x) = (
n
x
)pxq(n-μ)2] = Σ(x-μ)2x p(x)x) x= 0,1,2,…,n
p = the probability of success on any single trial
n = total number of trials
q = 1 -μ)2] = Σ(x-μ)2x p(x)p
x = number of successes in n trials.
Let μ and σ be the mean and standard deviation of the binomial random variable X. Instead of using the expectation summation rules to calculate μ and σ,
we can find μ and σ easily using the formulas μ= np, σ2= npq= np(1-μ)2] = Σ(x-μ)2x p(x)p),and σ=
√
npq
=
√
np(1−p)
Poisson probability model –
The probability density function of a Poisson random variable is
p
(
x
)
=λ
x
e
−λ
x !
x=0,1,2,…,
Both the mean and the variance of a Poisson random variable equals to λ, i.e., μ = λ and σ2= λ
Collection of Definitions:
Random Variable –A random variable is a rule that assigns one and only one numerical value to each sample point in a random experiment.
Discrete Random Variable – Discrete random variable is one kind of random variable that can assume values on countable number of points.
Continuous Random Variable – Continuous random variable is one kind of random variable that can assume values in one or more intervals.
Probability Distribution – The probability distribution of a discrete random variable is a graph, a table, or a formula that specifies the probability associated
with each possible value the random can assume.
Expectation of a Discrete Random Variable -μ)2] = Σ(x-μ)2x p(x)
The expectation of a discrete random variable is the population mean of this random variable. We can use the following formula to compute the expectation
of a discrete random variable μ=Σxp(x)=E(x).
Variance of Discrete Random Variable -μ)2] = Σ(x-μ)2x p(x) The variance of a discrete random variable is σ2= E[(x-μ)2] = Σ(x-μ)2x p(x)μ)2] = Σ(x-μ)2] = Σ(x-μ)2x p(x)μ)2p(x).
Standard Deviation of Discrete Random Variable -μ)2] = Σ(x-μ)2x p(x) The standard deviation of a discrete random variable is equal to the square root of the variance of this
random variable, i.e., σ=
√
σ2
Binomial Distribution -μ)2] = Σ(x-μ)2x p(x) The probability density function of a binomial random variable is P(X=x) = p(x) = (
n
x
)pxq(n-μ)2] = Σ(x-μ)2x p(x)x) , where
p = the probability of success on any single trial ;
n = total number of trials;
x = number of successes in n trials;
q = 1 -μ)2] = Σ(x-μ)2x p(x)p;
