Answer:
a) X can take the values {0,1,2,3,...,n}
The probability mass function of x is defined for k copies received correctly as:
![P(X=k)=\dbinom{n}{k}p^k(1-p)^{n-k}](https://tex.z-dn.net/?f=P%28X%3Dk%29%3D%5Cdbinom%7Bn%7D%7Bk%7Dp%5Ek%281-p%29%5E%7Bn-k%7D)
b) N should be N=2 to ensure that with probability 0.99 at least one copy of the message is received.
Step-by-step explanation:
The variable X:number of copies received correctly by the end user can be modeled as a binomial variable, as it is a sum of n Bernoulli variables with probability p.
X can take the values {0,1,2,3,...,n}
The probability mass function of x is defined for k copies received correctly as:
![P(X=k)=\dbinom{n}{k}p^k(1-p)^{n-k}](https://tex.z-dn.net/?f=P%28X%3Dk%29%3D%5Cdbinom%7Bn%7D%7Bk%7Dp%5Ek%281-p%29%5E%7Bn-k%7D)
being p: the probability that a packet is received correctly by an end user, and q=1-p: the probability that a packet is not received correctly by an end user.
b) We have to calculate n so as to have a probability P(x≥1)=0.99, when p=0.9.
![P(x\geq1)=1-P(x=0)=0.99\\\\\\P(x=0)=\dbinom{n}{0}p^0(1-p)^n=1*1*(1-p)^n=(1-0.9)^n=0.1^n\\\\\\P(x\geq1)=1-0.1^n=0.99\\\\0.1^n=1-0.99=0.01\\\\n\cdot ln(0.1)=ln(0.01)\\\\n=ln(0.01)/ln(0.1)=2](https://tex.z-dn.net/?f=P%28x%5Cgeq1%29%3D1-P%28x%3D0%29%3D0.99%5C%5C%5C%5C%5C%5CP%28x%3D0%29%3D%5Cdbinom%7Bn%7D%7B0%7Dp%5E0%281-p%29%5En%3D1%2A1%2A%281-p%29%5En%3D%281-0.9%29%5En%3D0.1%5En%5C%5C%5C%5C%5C%5CP%28x%5Cgeq1%29%3D1-0.1%5En%3D0.99%5C%5C%5C%5C0.1%5En%3D1-0.99%3D0.01%5C%5C%5C%5Cn%5Ccdot%20ln%280.1%29%3Dln%280.01%29%5C%5C%5C%5Cn%3Dln%280.01%29%2Fln%280.1%29%3D2)