Mathematics Homework Help

See attached

1. Let x1,x2,...xn$x_1,x_2,...x_n$ be iid random variables that follow the Poisson distribution with parameter λ$\lambda$, i.e. f(x;λ)=e−λλxx!$f(x;\lambda )=\frac{e^{-\lambda }{\lambda }^x}{x!}$.

1. (15 points) Find the methods of moments estimator for E[xi]=λ$E\left[x_i\right]=\lambda$.
2. (15 points) Is this estimate unbiased? (show your work)
3. (15 points) Write an R function that would yield Methods of moments estimate for sample mean of a random sample of iid Poisson variables.

• Argument: A vector of X$X$ values
• Output: MM estimate of E[xi]=λ$E\left[x_i\right]=\lambda$

1. (10 points) Use your function and x defined below to calculate the MM estimate of λ$\lambda$.

set.seed(1)
x=rpois(100, 3.6)

1. Let Xi$X_i$ be a random variable with parameter p$p$ and probability mass functionf(xi,p)=(nxi)pxi(1−p)n−xi$f(x_i,p)=(\frac{n}{x_i})p_i^x(1-p{)}^{n-x_i}$

1. (15 points) The maximum likelihood estimator for E[Xi]=np$E\left[X_i\right]=np$ is the sample mean. Is this estimate unbiased? (show your work)
2. (15 points) Write an R function that would yield ML estimate for E[Xi]$E\left[X_i\right]$.

• Argument: A vector of X$X$ values
• Output: ML estimate of E[xi]=np$E\left[x_i\right]=np$

1. (15 points) Use your function and y defined below to calculate the ML estimate of p$p$.

set.seed(1)
y=rbinom(n=30, size=12, p=0.85)
y

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