Question

4. (a) Let $ \left\{X_{n}\right\} $ be a sequence of i.i.d.uniform $ (0,1) $ random variables. Define $ Y=\min \left\{X_{n}\right\} $. - Show that $ Y \xrightarrow{D} 0 $ and $ Y \stackrel{P}{\rightarrow} 0 $. [40 Marks] (b) If $ T_{n}=X_{1}+X_{2}+\cdots+X_{n} $ is the sum of a random sample of size $ n $ with mean $ \mu $ and variance $ \sigma^{2} $, show that $ \frac{T_{n}-n \mu}{\sqrt{n} \sigma} \approx A N(0,1) $ for sufficiently large $ n $. [30 Marks] (c) Let $ X_{1}, X_{2}, \ldots \ldots, X_{30} $ be a random sample of size 30 from a Uniform $ [0,1] $ distribution. Find $ P\left(T_{30}<20\right) $ using the results in part (b). [30 marks]

          4. (a) Let \( \left\{X_{n}\right\} \) be a sequence of i.i.d.uniform \( (0,1) \) random variables. Define \( Y=\min \left\{X_{n}\right\} \).
- Show that \( Y \xrightarrow{D} 0 \) and \( Y \stackrel{P}{\rightarrow} 0 \).
[40 Marks]
(b) If \( T_{n}=X_{1}+X_{2}+\cdots+X_{n} \) is the sum of a random sample of size \( n \) with mean \( \mu \) and variance \( \sigma^{2} \), show that \( \frac{T_{n}-n \mu}{\sqrt{n} \sigma} \approx A N(0,1) \) for sufficiently large \( n \).
[30 Marks]
(c) Let \( X_{1}, X_{2}, \ldots \ldots, X_{30} \) be a random sample of size 30 from a Uniform \( [0,1] \) distribution. Find \( P\left(T_{30}<20\right) \) using the results in part (b).
[30 marks]

$4. (a) Let {Xn} be a sequence of i.i.d.uniform (0,1) random variables. Define Y=min{Xn}. - Show that Y D 0 and Y P→ 0. [40 Marks] (b) If Tn=X1+X2+⋯+Xn is the sum of a random sample of size n with mean μ and variance σ^2, show that (Tn-n μ)/(√(n)σ)≈ A N(0,1) for sufficiently large n. [30 Marks] (c) Let X1, X2, ……, X30 be a random sample of size 30 from a Uniform [0,1] distribution. Find P(T30<20) using the results in part (b). [30 marks]$

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