Transcript Normal approximation of Binomial probabilities

Normal approximation of
Binomial probabilities
Normal approximation of Binomial probabilities
 Recall binomial experiment:
 Identical trials
 Two outcomes: success and failure
 Probability for success and failure consistent
 Independent trials
Normal approximation of Binomial probabilities
 Binomial probability:
 Given n and p,
n k
nk
P( X  k )    p (1  p)
k
Normal approximation of Binomial probabilities
 When n is large, finding the above probability
becomes increasingly burdensome if k is in the
middle of n, since
n
k
P( X  k )  P ( X  k ) and
P( X  k )  P ( X  k ),

i 0
 Or
i
i
i

ik
i
i
i
P( X  k )  1  P( X  k )
 In this case, using complement may not help either.
Normal approximation of Binomial probabilities
 We have talked about using Poisson to approximate
binomial for n*p is less than 5. or n*p<5.
 Then how about n*p > 5?
 Here we can actually use normal distribution to
approximate binomial.
Normal approximation of Binomial probabilities
 How to approximate:
 Given any random variable X~BIN(n, p)
1. Find its mean (n*p) and variance (n*p*(1-p))
 2. Set  =n*p, and  2 = n*p*(1-p).
 3. Then we can consider X as X~N(n*p, n*p*(1-p) )
 4. To find probabilities, we can standardize X into Z and look them
up in the Z/Normal probability table.

Normal approximation of Binomial probabilities
 However, there is something we have missed.
 Previously, when we use Poisson to approximate
Binomial, they are both DISCRETE.
 But now, Binomial distribution is for DISCRETE
random variables and Normal distribution is for
CONTINUOUS random variables.
Normal approximation of Binomial probabilities
 What is the potential problem?
 Simply, how do we calculate P(X=k)?
 Recall that:
 For a discrete random variable, we just calculate it directly,
there is no problem.
 For a continuous random variable, we know that for ANY
INDIVIDUAL VALUE OF X, P(X=k)=0!!!
Normal approximation of Binomial probabilities
 We have to make some corrections.
 There is a technique called “continuity correction”.
 All we need to do is to add a “continuity correction
factor”.
 Under normal approximation,


P(X=k)=P(k-0.5 < X < k+0.5)
By this correction, we are computing the probability over an
interval instead at a single point.
Example
 A hotel has 100 rooms and the probability a room is
occupied on any given night is 0.6. Assume the
conditions of the binomial are met for the number of
occupied rooms on any given night.
Example
 1. Find the probability that there are 50 rooms
occupied at a given night using the exact
distribution.
 2. Find the probability that there are 50 rooms
occupied at a given night using normal
approximation.
Example
 3. Find the probability that there are at least 50
rooms are occupied at a given night using the exact
distribution.
 4. Find the probability that there are at least 50
rooms occupied at a given night using normal
approximation.