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4. In this problem, you will show that the Poisson random variable is a good approximation for the Binomial random variable in the limit using characteristic functions: (a) Find the characteristic function of the Binomial random variable. Hint: The Binomial theorem states the following: [ (a+b)^{n}=sum_{k=0}^{n}left(egin{array}{l} n \ k end{array} ight) a^{k} b^{n-k} ] (b) Find the characteristic function of the Poisson random variable. Hint: We know from Taylor series: [ e^{x}=sum_{k=0}^{infty} frac{x^{k}}{k !} ] (c) As ( n ) approaches ( infty ), denote ( n p ) as ( lambda ) and use the fact that [ lim _{n ightarrow infty}left(1+frac{x}{n} ight)^{n}=e^{x} ] to prove the question. (d) What can you say about the value of ( n, p ), and ( lambda ) at this point? When is the Poisson random variable a good approximation for the Binomial random variable? Hint: ( lambda ) is the mean of the Poisson distribution: ( mathbb{E}[X]=lambda ) where ( X sim operatorname{Poisson}(lambda) ). (e) Haemophilia is a rare genetic bleeding disorder that affects 1 in every 10,000 people in the world, where the person's blood does not clot by itself. In a sample of 2,000 people, what is the probability that exactly 1 person has haemophilia? Solve this question using the Binomial RV as well as its Poisson approximation and compare.

          4. In this problem, you will show that the Poisson random variable is a good approximation for the Binomial random variable in the limit using characteristic functions:
(a) Find the characteristic function of the Binomial random variable.
Hint: The Binomial theorem states the following:
[
(a+b)^{n}=sum_{k=0}^{n}left(egin{array}{l}
n \
k
end{array}
ight) a^{k} b^{n-k}
]
(b) Find the characteristic function of the Poisson random variable.
Hint: We know from Taylor series:
[
e^{x}=sum_{k=0}^{infty} frac{x^{k}}{k !}
]
(c) As ( n ) approaches ( infty ), denote ( n p ) as ( lambda ) and use the fact that
[
lim _{n 
ightarrow infty}left(1+frac{x}{n}
ight)^{n}=e^{x}
]
to prove the question.
(d) What can you say about the value of ( n, p ), and ( lambda ) at this point? When is the Poisson random variable a good approximation for the Binomial random variable?
Hint: ( lambda ) is the mean of the Poisson distribution: ( mathbb{E}[X]=lambda ) where ( X sim operatorname{Poisson}(lambda) ).
(e) Haemophilia is a rare genetic bleeding disorder that affects 1 in every 10,000 people in the world, where the person's blood does not clot by itself. In a sample of 2,000 people, what is the probability that exactly 1 person has haemophilia? Solve this question using the Binomial RV as well as its Poisson approximation and compare.

$4. In this problem, you will show that the Poisson random variable is a good approximation for the Binomial random variable in the limit using characteristic functions: (a) Find the characteristic function of the Binomial random variable. Hint: The Binomial theorem states the following: [ (a+b)^n=sumk=0^nleft(eginarrayl n k endarray ight) a^k b^n-k ] (b) Find the characteristic function of the Poisson random variable. Hint: We know from Taylor series: [ e^x=sumk=0^infty fracx^kk ! ] (c) As ( n ) approaches ( infty ), denote ( n p ) as ( lambda ) and use the fact that [ lim n ightarrow inftyleft(1+fracxn ight)^n=e^x ] to prove the question. (d) What can you say about the value of ( n, p ), and ( lambda ) at this point? When is the Poisson random variable a good approximation for the Binomial random variable? Hint: ( lambda ) is the mean of the Poisson distribution: ( mathbbE[X]=lambda ) where ( X sim operatornamePoisson(lambda) ). (e) Haemophilia is a rare genetic bleeding disorder that affects 1 in every 10,000 people in the world, where the person's blood does not clot by itself. In a sample of 2,000 people, what is the probability that exactly 1 person has haemophilia? Solve this question using the Binomial RV as well as its Poisson approximation and compare.$

Added by 7Gygu I.

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