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Class Note
Feb 13, 2003
1. Definitions and theory for binomial r.v’s.
Know what a binomial experiment is. (e.g. toss a die 10 times)
If X is a binomial random variable
▫ representing the number of successes
▫ in n independent, identical trials,
▫ with probability of success p remaining constant from trial to
trial,
then is called a binomial r.v. with parameters n and p and
we write
X~ Bin (n, p).
e.g. X=the number of times out of ten that you get 4 or 6, thus,
X~Bin(10,1/3).
- The probability density function (pdf) of X~Bin(n,p) is
p(k) = P(X=k) = n!/[k!(n-k)!] pk (1-p)n-k , for k=0,…,n.
You can get these probabilities from the tables I gave you. Once you
specify n
and p, you can find P(X=0), P(X=1), …, P(X=n).
- The cumulative density function (cdf) of X~Bin(n,p) is P(X ≤ k), for
all k.
For k integer between 0 and n we have that
P(X < k) = P(X ≤ k-1)
Note that is not true for discrete random variables in general!!!
Binomial
random variables can take only the integer values 0,1,…,n (since is
the number
of successes out of n). If X is not a binomial variable it might be the
case that
the possible values of X are 2, 2.5, 3, 3.5 and 4. Then, in this case
P(X < 3) = P(X ≤ 2.5).
- For a binomial random variable X~ Bin(np)
1.
μ = E(X) = np
2.
σ2 = Var (X)= np(1-p)
So…for binomial random variables you do not need to use the
general formulas
for expected value and variance!
 Continuous Random Variables
1. Definitions and theory for continuous r.v’s in general.
- X is a continuous r.v. if it can take any possible value in a interval or
intervals.
-
-
-
P(X=x) = 0 for and real number x, thus we need different
definition for the pdf.
The pdf of a X is a curve, f(x), such that
▫ f(x) ≥ 0 for and real number x.
▫ the area under the curve and above the x-axis is equal to 1.
▫ for and real numbers a and b, P(a ≤ X ≤ b) is given by the area
under the
curve and above the interval.
The cdf is defined similarly to the case of discrete r.v. and it
is denoted by F(x).
Note that since for continuous r.v. P(X=x) = 0,
F(x) = P(X ≤ x) = P(X < x)
Therefore, F(x) is the area under the curve f(x) above the interval (-∞,
x).
(i.e. the area under the curve for all the values less than x).
2. Definitions and theory for Normal r.v’s.
- If a r.v. X has a normal distribution with mean μ, and standard
deviation σ we
write X~ N (μ, σ2).
- Knowing μ and σ specifies the particular normal distribution out of
the class of
all normal distributions. (Similarly, knowing n and p specifies a
particular
binomial distribution.)
-
The pdf of any normal r.v X, called normal curve, is
▫ Symmetric
▫ Bell shaped
▫ Centered at the mean, μ.
The normal curve also satisfies the three rules listed above for any
pdf of a
continuous r.v.
-
The standard normal random variable has mean 0 and
standard deviation 1.
We denote it with Z, and we have that Z~ N(0,1).
We have the tables for all the probabilities of the form P(Z ≤
z), for Z~ N(0,1).
So for any X~ N(μ, σ2), we can obtain any probabilities of interest
using the
following “standardization theorem’.
-
-
If X~ N(μ, σ2), then {(X- μ)/ σ } ~ N(0,1), thus,
P(X ≤ x) = P [(X- μ)/ σ ≤ (x- μ)/ σ] = P[Z ≤ (x- μ)/ σ] = P(Z ≤
z),
Where z = (x- μ)/ σ, is called the z-score of x.
-
Now, how to find cumulative probabilities of X using the
tables!
▫ First find the z-score of x (or x’s if more than one) to be able to use
the tables.
▫ Think what is the area under the curve that corresponds to this
probability.
▫ Figure out how you can get this probability using probabilities
rules and
values form the tables.
▫ Have in mind that the normal curve is symmetric and that the total
area under
the curve is equal to 1.
-
The empirical rule for the standard deviation on page 44 is
valid for all bell
shaped distributions (μ ± σ, μ ± 2σ, μ ± 3σ, approximate intervals),
but it is
EXACTLY RIGHT in the case of normal distribution.
-
Know how to find percentiles, i.e. for X~ N(μ, σ2), i.e. know
how to find x0 such
that P( X ≤ x0) = α where α is a known probability.
e.g. Z~N(0,1), P(Z≤ z0) = 0.95, then z0 = 1.64.
-
Know how to use the normal approximation for binomial
distribution
probabilities, when the number of trials is large.