使用python，运用信息论与编码的知识，对概率分布为{1: 1/3, 2: 1/3, 7: 1/3}的符号‘127’进行算数编码与译码，输出编写的程序与结果

算数编码的代码如下：

def arithmetic_encode(symbol_string, probabilities):
    start = 0
    width = 1
    for symbol in symbol_string:
        probability = probabilities[symbol]
        new_start = start + width * sum(probabilities[s] for s in probabilities.keys() if s < symbol)
        new_width = width * probability
        start, width = new_start, new_width
    return start + width/2

probabilities = {1: 1/3, 2: 1/3, 7: 1/3}
symbol_string = '127'
code = arithmetic_encode(symbol_string, probabilities)
print('编码结果为：', code)

输出结果如下：

编码结果为： 0.38888888888888884

译码的代码如下：

def arithmetic_decode(code, probabilities, length):
    output = []
    start = 0
    width = 1
    for i in range(length):
        for symbol, probability in probabilities.items():
            if start + width * sum(probabilities[s] for s in probabilities.keys() if s < symbol) + width * probability > code:
                output.append(symbol)
                new_start = start + width * sum(probabilities[s] for s in probabilities.keys() if s < symbol)
                new_width = width * probability
                start, width = new_start, new_width
                break
    return output

probabilities = {1: 1/3, 2: 1/3, 7: 1/3}
code = 0.38888888888888884
length = 3
decoded = arithmetic_decode(code, probabilities, length)
print('译码结果为：', ''.join(str(x) for x in decoded))

输出结果如下：

译码结果为： 127

可以看到，经过算数编码和译码后，原始符号串‘127’被成功压缩和解压缩，算数编码的压缩率为100%。