For large value of n, and moderate values of the probabiliy of success p (roughly, 0.05 Leftarrow p Leftarrow 0.95), the binomial distribution can be

glamrockqueen7 2021-03-18 Answered
For large value of n, and moderate values of the probabiliy of success p (roughly, 0.05  p  0.95), the binomial distribution can be approximated as a normal distribution with expectation mu = np and standard deviation
σ=np(1  p). Explain this approximation making use the Central Limit Theorem.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Asma Vang
Answered 2021-03-19 Author has 93 answers

Step 1
Alternatively prove that:
Theorem:
If x is a random variable with distribution B(n, p), then for sufficiently large n, the distribution of the
variablez =x  μσ  N(0, 1)
where μ=np σ2=np(1  p)
Proof:
It can be prove using Moment generating function for binomial distribution. It's given as,
Mx(θ)=(q + peθ)n
where q=1  p.
Step 2
By the linear transformation properties of the moment generating function.
Mz(θ)=e  μ θσMx(θσ)=e  μ θσ(q + pe θσ)n
Taking the natural log of both sides, and then expanding the power series of eθ/σ
Then, ln Mz(θ)=  μ θσ + n ln(q + pe θσ)=  μ θσ + n ln(q + p k=11k!(θσ)k)
Since p + q=1.
=  μ θσ + n ln(1 + p k=11k!(θσ)k)
If n is made sufficiently large σ=npq can be made large enough that for any fixed theta the absolute value of the sum above will be less than 1.
Let t=pm=11m!(θσ)m
Thus for sufficiently large n, |t| < 1.
The ln term in the previous expression is ln(1 + t) where |t| < 1, and so we may expand this term as follows:
ln(1 + t)=k=1(1)k  1tkk
Step 3
This means that
ln Mz(θ)=  μ θσ + n ln(1 + t)=  μ θσ + n k=1(1)k1

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more