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If you are trying to prove the independence of $\overline X$ and $S^2$ via change of variables, suggest you using an orthogonal transformation for ease of calculation. The result can be proved without finding the joint pdf of $(\overline X,S^2)$ and showing they are independent because the joint pdf factors as the product of two marginals.

Consider the transformation $$(X_1,X_2,\ldots,X_n)\to(Y_1,Y_2,\ldots,Y_n)$$ such that $$\begin{pmatrix}Y_1\Y_2\\vdots\Y_n\end{pmatrix}=Q\begin{pmatrix}X_1\X_2\\vdots\X_n\end{pmatrix}$$

, where $Q$ is an $n\times n$ orthogonal matrix with the first row $$\left(\frac{1}{\sqrt{n}},\frac{1}{\sqrt{n}},\ldots,\frac{1}{\sqrt{n}}\right)$$

Then, $$Y_1=\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i=\sqrt{n}\overline X \quad\text{ and }\quad\sum_{i=1}^n Y_i^2=\sum_{i=1}^n X_i^2$$

Clearly, $$(X_1,X_2,\ldots,X_n)\in\mathbb R^n\implies (Y_1,Y_2,\ldots,Y_n)\in\mathbb R^n $$

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import 'package:flutter/material.dart';

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {

  @override

  Widget build(BuildContext context) {

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void main() {
  print("hi");

  dynamic x = 1;

  print(x);

  x = "hi 2";

  print(x);

Divyosmi

from tensorflow import keras
model = keras.model.load_model("./model.h5")
age = int(input("age: input age(only numbers) "))
sex = int(input("sex: input 1 for female and 0 for male(only numbers "))
cp = int(input("cp: input 0 for Asymptomatic, 1 for Atypical Angina, 2 for Nonanginal Pain, 3 for Typical Angina(only numbers) "))
trestbps = int(input("trestbps: input trestbps(only numbers) "))
chol = int(input("chol: input cholesterol(only numbers) "))
fbs= int(input("fbs: input 1 for True and 0 for False(only numbers "))
restecg=int(input("restecg: input 0 for Having ST-T wave abnormal, 1 for Hypertrophy, 2 for Normal(only numbers) "))
thalach = int(input("thalach: input thalach(only numbers) "))

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import datetime

welcome = "welcome to Urja Flower Wallet"

import hashlib

import time

class Block:

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import hashlib

import time

class Block:

    def __init__(self, index, proof_no, prev_hash, data, timestamp=None):

        self.index = index

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{\"class_name\": \"Sequential\", \"config\": {\"name\": \"sequential\", \"layers\": [{\"class_name\": \"Model\", \"config\": {\"name\": \"vgg16\", \"layers\": [{\"class_name\": \"InputLayer\", \"config\": {\"batch_input_shape\": [null, 512, 512, 3], \"dtype\": \"float32\", \"sparse\": false, \"ragged\": false, \"name\": \"input_1\"}, \"name\": \"input_1\", \"inbound_nodes\": []}, {\"class_name\": \"Conv2D\", \"config\": {\"name\": \"block1_conv1\", \"trainable\": false, \"dtype\": \"float32\", \"filters\": 64, \"kernel_size\": [3, 3], \"strides\": [1, 1], \"padding\": \"same\", \"data_format\": \"channels_last\", \"dilation_rate\": [1, 1], \"activation\": \"relu\", \"use_bias\": true, \"kernel_initializer\": {\"class_name\": \"GlorotUniform\", \"config\": {\"seed\": null}}, \"bias_initializer\": {\"class_name\": \"Zeros\", \"config\": {}}, \"kernel_regularizer\": null, \"bias_regularizer\": null, \"activity_regularizer\": null, \"kernel_constraint\": null, \"bias_constraint\": null}, \"name\": \"block1_conv1\

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"""VGG16 model for Keras.

# Reference

- [Very Deep Convolutional Networks for Large-Scale Image Recognition](
    https://arxiv.org/abs/1409.1556) (ICLR 2015)

"""
from __future__ import absolute_import
from __future__ import division