Questions

LLN

Let there be i.i.d. r.v.s. X1,X2,XnX_1, X_2, \ldots X_n with mean μ\mu and variance σ2\sigma^2.

Give a value of nn (a specific number) that will ensure that there is at least a 99% chance that the sample mean will be within 2 standard deviations of the true mean μ\mu.

Gamma CLT

Explain why a Gamma random variable with parameters (n,λ)(n, \lambda) is approximately Normal when n is large.

Let XnGamma(n,λ)X_n \sim Gamma(n, \lambda). Determine aa and bb such that XnabN(0,1)\frac{X_n - a}{b} \to N(0,1) as nn \to \infty

Markov Chain

Suppose XnX_n is a two-state Markov chain with transition matrix

Q=(1ααβ1β)Q = \left(\begin{array}{cc} 1-\alpha & \alpha \\ \beta & 1-\beta \end{array}\right)

The rows and columns are indexed 0,1 such that q0,0=1α,q0,1=α,q1,0=β,q1,1=1βq_{0,0}=1-\alpha, q_{0,1}=\alpha, q_{1,0}=\beta, q_{1,1}=1-\beta.

Find the stationary distribution s=(s0,s1)\vec{s} = (s_0, s_1) of XnX_n by solving sQ=s\vec{s} Q = \vec{s}.

Show that this Markov Chain is reversible under the stationary distribution found in the previous question.

Let Zn=(Xn1,Xn)Z_n = (X_{n-1}, X_n). Is ZnZ_n a Markov chain? If so, what are the states and transition matrix?

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