Ian Chen 發問時間： 科學數學 · 8 年前

# 我有統計學問題~急><

prove by induction by

p(E1∪ E2∪ ...∪ En)≦上nΣ下i=1p(Ei)

for any finite sequence of events E1,E2,...,and En

### 1 個解答

• yc
Lv 5
8 年前
最佳解答

Let S(n) be the statement "p(E1∪ E2∪ ...∪ En)≦Σ(i=1→n)p(Ei) where n is a positive integer.

When n=1, p(E1∪ E2∪ ...∪ En)=p(E1), Σ(i=1→n)p(Ei)=p(E1).

∴S(1) is true.

Assuming S(k) is true for some positive integer k,

i.e. p(E1∪ E2∪ ...∪ Ek)≦Σ(i=1→k)p(Ei) .

When n=k+1,

p(E1∪ E2∪ ...∪ Ek+1)

=p[(E1∪ E2∪ ...∪ Ek)∪ Ek+1]

=p(E1∪ E2∪ ...∪ Ek)＋p(Ek+1)－p[(E1∪ E2∪ ...∪ Ek)∩ Ek+1]

≦p(E1∪ E2∪ ...∪ Ek)＋p(Ek+1)

=Σ(i=1→k)p(Ei)＋p(Ek+1)

=Σ(i=1→k+1)p(Ei)

∴S(k+1) is true if S(k) is true.

By principle of MI, S(n) is true for all positive integers n.
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://tw.answers.yahoo.com/question/index?qid=20120913000015KK01804", "fetch_time": 1601484654000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2020-40/segments/1600402127075.68/warc/CC-MAIN-20200930141310-20200930171310-00310.warc.gz", "warc_record_offset": 632181783, "warc_record_length": 26342, "token_count": 395, "char_count": 728, "score": 3.625, "int_score": 4, "crawl": "CC-MAIN-2020-40", "snapshot_type": "latest", "language": "en", "language_score": 0.46855199337005615, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00031-of-00064.parquet"}