4.3: Thamani ya Maana au Inatarajiwa na Kupotoka kwa kiwango
- Page ID
- 180995
Thamani inayotarajiwa mara nyingi hujulikana kama wastani wa “muda mrefu” au maana. Hii ina maana kwamba kwa muda mrefu wa kufanya jaribio mara kwa mara, ungependa kutarajia wastani huu.
Unatupa sarafu na kurekodi matokeo. Je! Ni uwezekano gani kwamba matokeo ni vichwa? Kama flip sarafu mara mbili, je, uwezekano kukuambia kwamba flips hizi kusababisha vichwa moja na mkia mmoja? Unaweza toss sarafu ya haki mara kumi na kurekodi vichwa tisa. Kama ulivyojifunza katika Sura ya 3, uwezekano hauelezei matokeo ya muda mfupi ya jaribio. Inatoa taarifa kuhusu kile kinachoweza kutarajiwa kwa muda mrefu. Ili kuonyesha hili, Karl Pearson mara moja alipoteza sarafu ya haki mara 24,000! Alirekodi matokeo ya kila toss, akipata vichwa mara 12,012. Katika majaribio yake, Pearson alionyesha Sheria ya Hesabu Kubwa.
Sheria ya Idadi kubwa inasema kwamba, kama idadi ya majaribio katika jaribio la uwezekano huongezeka, tofauti kati ya uwezekano wa kinadharia wa tukio na mzunguko wa jamaa unakaribia sifuri (uwezekano wa kinadharia na mzunguko wa jamaa hupata karibu na karibu pamoja). Wakati wa kutathmini matokeo ya muda mrefu ya majaribio ya takwimu, mara nyingi tunataka kujua matokeo ya “wastani”. Hii “wastani wa muda mrefu” inajulikana kama thamani ya maana au inatarajiwa ya jaribio na inaashiria kwa barua ya Kigiriki\(\mu\). Kwa maneno mengine, baada ya kufanya majaribio mengi ya jaribio, ungependa kutarajia thamani hii ya wastani.
Ili kupata thamani inatarajiwa au wastani wa muda mrefu,\(\mu\), tu kuzidisha kila thamani ya variable random na uwezekano wake na kuongeza bidhaa.
Mfano\(\PageIndex{1}\)
Timu ya soka ya wanaume ina soka sifuri, moja, au siku mbili kwa wiki. uwezekano kwamba wao kucheza siku zero ni 0.2, uwezekano kwamba wao kucheza siku moja ni 0.5, na uwezekano kwamba wao kucheza siku mbili ni 0.3. Kupata wastani wa muda mrefu au inatarajiwa thamani\(\mu\),, ya idadi ya siku kwa wiki timu ya soka ya wanaume ina soka.
Suluhisho
Kwa kufanya tatizo, kwanza basi kutofautiana random idadi\(X =\) ya siku timu ya soka ya wanaume ina soka kwa wiki. \(X\)inachukua maadili 0, 1, 2. Kujenga meza PDF kuongeza safu\(x*P(x)\). Katika safu hii, utazidisha kila\(x\) thamani kwa uwezekano wake.
| \(x\) | \(P(x)\) | \(x*P(x)\) |
|---|---|---|
| \ (x\) "> 0 | \ (P (x)\) "> 0.2 | \ (X* p (x)\) "> (0) (0.2) = 0 |
| \ (x\) ">1 | \ (P (x)\) "> 0.5 | \ (X* p (x)\) "> (1) (0.5) = 0.5 |
| \ (x\) "> 2 | \ (P (x)\) "> 0.3 | \ (X* p (x)\) "> (2) (0.3) = 0.6 |
Ongeza safu ya mwisho\(x*P(x)\) ili kupata wastani wa muda mrefu au thamani inayotarajiwa:
\[(0)(0.2) + (1)(0.5) + (2)(0.3) = 0 + 0.5 + 0.6 = 1.1. \nonumber\]
Thamani inayotarajiwa ni 1.1. Timu ya soka ya wanaume ingekuwa, kwa wastani, inatarajia kucheza soka siku 1.1 kwa wiki. Nambari 1.1 ni wastani wa muda mrefu au thamani inayotarajiwa ikiwa timu ya soka ya wanaume inacheza soka wiki baada ya wiki baada ya wiki baada ya wiki. Tunasema\(\mu = 1.1\).
Mfano\(\PageIndex{2}\)
Pata thamani inayotarajiwa ya idadi ya mara kilio cha mtoto wachanga kinachomfufua mama yake baada ya usiku wa manane. Thamani inayotarajiwa ni idadi inayotarajiwa ya mara kwa wiki kilio cha mtoto wachanga kinachomfufua mama yake baada ya usiku wa manane. Tumia kupotoka kwa kiwango cha kutofautiana pia.
| \(x\) | \(P(x)\) | \(x*P(x)\) | (x -\(\mu)^{2} ⋅ P(x)\) |
|---|---|---|---|
| \ (x\)” style="wima align:katikati; "> 0 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 0) = \dfrac{2}{50}\) | \ (X*p (x)\)” style="wima align:katikati; ">\((0)\left(\dfrac{2}{50}\right) = 0\) | \ (\ mu) ^ {2} 合 P (x)\)” style="wima align:katikati; "> (0 - 2.1) 2 合 0.04 = 0.1764 |
| \ (x\)” style="wima align:katikati; "> 1 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 1) = \dfrac{11}{50}\) | \ (X*p (x)\)” style="wima align:katikati; ">\((1)\left(\dfrac{11}{50}\right) = \dfrac{11}{50}\) | \ (\ mu) ^ {2} 合 P (x)\)” style="wima align:katikati; "> (1 - 2.1) 2 合 0.22 = 0.2662 |
| \ (x\)” style="wima align:katikati; "> 2 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 2) = \dfrac{23}{50}\) | \ (X*p (x)\)” style="wima align:katikati; ">\((2)\left(\dfrac{23}{50}\right) = \dfrac{46}{50}\) | \ (\ mu) ^ {2} 合 P (x)\)” style="wima align:katikati; "> (2 - 2.1) 2 合 0.46 = 0.0046 |
| \ (x\)” style="wima align:katikati; "> 3 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 3) = \dfrac{9}{50}\) | \ (X*p (x)\)” style="wima align:katikati; ">\((3)\left(\dfrac{9}{50}\right) = \dfrac{27}{50}\) | \ (\ mu) ^ {2} 合 P (x)\)” style="wima align:katikati; "> (3 — 2.1) 2 合 0.18 = 0.1458 |
| \ (x\)” style="wima align:katikati; "> 4 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 4) = \dfrac{4}{50}\) | \ (X*p (x)\)” style="wima align:katikati; ">\((4)\left(\dfrac{4}{50}\right) = \dfrac{16}{50}\) | \ (\ mu) ^ {2} 合 P (x)\)” style="wima align:katikati; "> (4 - 2.1) 2 合 0.08 = 0.2888 |
| \ (x\)” style="wima align:katikati; "> 5 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 5) = \dfrac{1}{50}\) | \ (X*p (x)\)” style="wima align:katikati; ">\((5)\left(\dfrac{1}{50}\right) = \dfrac{5}{50}\) | \ (\ mu) ^ {2} 合 P (x)\)” style="wima align:katikati; "> (5 — 2.1) 2 合 0.02 = 0.1682 |
Ongeza maadili katika safu ya tatu ya meza ili kupata thamani inayotarajiwa ya\(X\):
\[\mu = \text{Expected Value} = \dfrac{105}{50} = 2.1 \nonumber\]
Tumia\(\mu\) ili kukamilisha meza. Safu ya nne ya meza hii itatoa maadili unayohitaji kuhesabu kupotoka kwa kawaida. Kwa kila thamani\(x\), kuzidisha mraba wa kupotoka kwake kwa uwezekano wake. (Kila kupotoka ina muundo\(x – \mu\).
Ongeza maadili katika safu ya nne ya meza:
\[0.1764 + 0.2662 + 0.0046 + 0.1458 + 0.2888 + 0.1682 = 1.05 \nonumber\]
Mkengeuko wa kawaida\(X\) ni mizizi ya mraba ya jumla hii:\(\sigma = \sqrt{1.05} \approx 1.0247\)
Maana, μ, ya kazi ya uwezekano wa kipekee ni thamani inayotarajiwa.
\[μ=∑(x∙P(x))\nonumber\]
Kupotoka kwa kiwango, Σ, ya PDF ni mizizi ya mraba ya ugomvi.
\[σ=\sqrt{∑[(x – μ)2 ∙ P(x)]}\nonumber\]
Wakati matokeo yote katika usambazaji uwezekano ni sawa, hizi formula sanjari na wastani na kiwango kupotoka ya seti ya matokeo iwezekanavyo.
Zoezi\(\PageIndex{2}\)
Mtafiti wa hospitali anavutiwa na idadi ya mara mgonjwa wa wastani baada ya op atapiga muuguzi wakati wa mabadiliko ya saa 12. Kwa sampuli ya random ya wagonjwa 50, habari zifuatazo zilipatikana. Thamani inatarajiwa ni nini?
| \(x\) | \(P(x)\) |
|---|---|
| \ (x\)” style="wima align:katikati; "> 0 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 0) = \dfrac{4}{50}\) |
| \ (x\)” style="wima align:katikati; "> 1 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 1) = \dfrac{8}{50}\) |
| \ (x\)” style="wima align:katikati; "> 2 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 2) = \dfrac{16}{50}\) |
| \ (x\)” style="wima align:katikati; "> 3 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 3) = \dfrac{14}{50}\) |
| \ (x\)” style="wima align:katikati; "> 4 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 4) = \dfrac{6}{50}\) |
| \ (x\)” style="wima align:katikati; "> 5 | \ (P (x)\)” style="wima align:katikati; ">\(P(x = 5) = \dfrac{2}{50}\) |
- Jibu
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Thamani inayotarajiwa ni 2.24
\[(0)\dfrac{4}{50} + (1)\dfrac{8}{50} + (2)\dfrac{16}{50} + (3)\dfrac{14}{50} + (4)\dfrac{6}{50} + (5)\dfrac{2}{50} = 0 + \dfrac{8}{50} + \dfrac{32}{50} + \dfrac{42}{50} + \dfrac{24}{50} + \dfrac{10}{50} = \dfrac{116}{50} = 2.32\]
Mfano\(\PageIndex{2}\)
Tuseme unacheza mchezo wa nafasi ambayo namba tano huchaguliwa kutoka 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Kompyuta nasibu huchagua namba tano kutoka sifuri hadi tisa na uingizwaji. Unalipa $2 kucheza na inaweza kufaidika $100,000 ikiwa unafanana na namba zote tano ili (unapata $2 nyuma pamoja na $100,000). Kwa muda mrefu, faida yako inatarajiwa ya kucheza mchezo ni nini?
Ili kufanya tatizo hili, weka meza ya thamani inayotarajiwa kwa kiasi cha fedha unachoweza kufaidika.
Hebu kiasi\(X =\) cha fedha unachofaidika. Maadili ya\(x\) si 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Kwa kuwa una nia ya faida yako (au hasara), maadili ya\(x\) ni dola 100,000 na dola -2.
kushinda, lazima kupata namba zote tano sahihi, ili. Uwezekano wa kuchagua namba moja sahihi ni\(\dfrac{1}{10}\) kwa sababu kuna namba kumi. Unaweza kuchagua idadi zaidi ya mara moja. Uwezekano wa kuchagua namba zote tano kwa usahihi na kwa utaratibu ni
\[\begin{align*} \left(\dfrac{1}{10}\right) \left(\dfrac{1}{10}\right) \left(\dfrac{1}{10}\right) \left(\dfrac{1}{10}\right) \left(\dfrac{1}{10}\right) &= (1)(10^{-5}) \\[5pt] &= 0.00001. \end{align*}\]
Kwa hiyo, uwezekano wa kushinda ni 0.00001 na uwezekano wa kupoteza ni
\[1−0.00001=0.99999.1−0.00001 = 0.99999.\nonumber\]
Jedwali la thamani linalotarajiwa ni kama ifuatavyo:
| \(x\) | \(P(x)\) | \(xP(x)\) | |
|---|---|---|---|
| Hasara | \ (x\) ">—2 | \ (P (x)\) "> 0.99999 | \ (XP (x)\) "> (—2) (0.99999) = —1.99998 |
| Faida | \ (x\) "> 100,000 | \ (P (x)\) "> 0.00001 | \ (XP (x)\) "> (100000) (0.00001) = 1 |
Tangu -0.99998 ni kuhusu -1, ungependa, kwa wastani, wanatarajia kupoteza takriban $1 kwa kila mchezo kucheza. Hata hivyo, kila wakati kucheza, ama kupoteza $2 au faida $100,000. the $1 ni wastani au inatarajiwa LOSS kwa kila mchezo baada ya kucheza mchezo huu tena na tena.
Zoezi\(\PageIndex{3}\)
Unacheza mchezo wa nafasi ambayo kadi nne hutolewa kutoka kwenye staha ya kawaida ya kadi 52. Unadhani suti ya kila kadi kabla ya inayotolewa. Kadi zimebadilishwa kwenye staha kwenye kila kuteka. Unalipa $1 kucheza. Ikiwa unadhani suti sahihi kila wakati, unapata pesa yako nyuma na $256. Ni faida yako inatarajiwa ya kucheza mchezo juu ya muda mrefu nini?
- Jibu
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Hebu kiasi\(X =\) cha fedha unachofaidika. \(x\)Thamani -ni - $1 na $256.
Uwezekano wa kubadili suti sahihi kila wakati ni\(\left(\dfrac{1}{4}\right) \left(\dfrac{1}{4}\right) \left(\dfrac{1}{4}\right) \left(\dfrac{1}{4}\right) = \dfrac{1}{256} = 0.0039\)
Uwezekano wa kupoteza ni\(1 – \dfrac{1}{256} = \dfrac{255}{256} = 0.9961\)
\((0.0039)256 + (0.9961)(–1) = 0.9984 + (–0.9961) = 0.0023\)au\(0.23\) senti.
Mfano\(\PageIndex{4}\)
Tuseme wewe kucheza mchezo na sarafu upendeleo. Kucheza kila mchezo na tossing sarafu mara moja. \(P(\text{heads}) = \dfrac{2}{3}\)na\(P(\text{tails}) = \dfrac{1}{3}\). Kama toss kichwa, kulipa $6. Kama toss mkia, wewe kushinda $10. Kama wewe kucheza mchezo huu mara nyingi, wewe kuja nje mbele?
- Eleza kutofautiana kwa random\(X\).
- Jaza zifuatazo inatarajiwa thamani meza.
- ni thamani inatarajiwa\(\mu\) nini,? Je! Unatoka mbele?
Solutions
a.
\(X\)= kiasi cha faida
| \(x\) | ____ | ____ | |
|---|---|---|---|
| KUSHINDA | \ (x\)” class="lt-stats-739">10 | \(\dfrac{1}{3}\) | ____ |
| KUPOTEZA | \ (x\)” class="lt-stats-739">____ | ____ | \(\dfrac{-12}{3}\) |
b.
| \(x\) | \(P(x)\) | \(xP(x)\) | |
|---|---|---|---|
| KUSHINDA | \ (x\)” class="lt-stats-739">10 | \ (P (x)\)” class="lt-stats-739">\(\dfrac{1}{3}\) | \ (XP (x)\)” class="lt-stats-739">\(\dfrac{10}{3}\) |
| KUPOTEZA | \ (x\)” class="lt-stats-739">—6 | \ (P (x)\)” class="lt-stats-739">\(\dfrac{2}{3}\) | \ (XP (x)\)” class="lt-stats-739">\(\dfrac{-12}{3}\) |
c.
Ongeza safu ya mwisho ya meza. thamani inatarajiwa\(\mu = \dfrac{-2}{3}\). Kupoteza, kwa wastani, kuhusu 67 senti kila wakati kucheza mchezo hivyo si kuja nje mbele.
Zoezi\(\PageIndex{4}\)
Tuseme wewe kucheza mchezo na spinner. Kucheza kila mchezo na inazunguka spinner mara moja. \(P(\text{red}) = \dfrac{2}{5}\),\(P(\text{blue}) = \dfrac{2}{5}\), na\(P(\text{green}) = \dfrac{1}{5}\). Kama nchi juu ya nyekundu, kulipa $10. Kama nchi juu ya bluu, huna kulipa au kushinda kitu chochote. Kama nchi juu ya kijani, kushinda $10. Jaza zifuatazo inatarajiwa thamani meza.
| \(x\) | \(P(x)\) | ||
|---|---|---|---|
| Nyekundu | \ (x\) "> | \ (P (x)\) "> | \(-\dfrac{20}{5}\) |
| Bluu | \ (x\) "> | \ (P (x)\) ">\(\dfrac{2}{5}\) | |
| Green | \ (x\) ">10 | \ (P (x)\) "> |
- Jibu
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\(x\) \(P(x)\) \(x*P(x)\) Nyekundu \ (x\) ">—10 \ (P (x)\) ">\(\dfrac{2}{5}\) \ (X*p (x)\) ">\(-\dfrac{20}{5}\) Bluu \ (x\) "> 0 \ (P (x)\) ">\(\dfrac{2}{5}\) \ (X*p (x)\) ">\(\dfrac{0}{5}\) Green \ (x\) ">10 \ (P (x)\) ">\(\dfrac{1}{5}\) \ (X*p (x)\) ">\(\dfrac{1}{5}\)
Kama data, mgawanyo wa uwezekano una upungufu wa kawaida. Ili kuhesabu kiwango cha kupotoka (σ) cha usambazaji wa uwezekano, pata kila kupotoka kutoka kwa thamani yake inayotarajiwa, mraba, uizidishe kwa uwezekano wake, kuongeza bidhaa, na kuchukua mizizi ya mraba. Ili kuelewa jinsi ya kufanya hesabu, angalia meza kwa idadi ya siku kwa wiki timu ya soka ya wanaume ina soka. Ili kupata kupotoka kwa kawaida, ongeza viingilio kwenye safu iliyoandikwa\((x) – \mu^{2}P(x)\) na uchukue mizizi ya mraba.
| \(x\) | \(P(x)\) | \(x*P(x)\) | \((x – \mu)^{2}P(x)\) |
|---|---|---|---|
| \ (x\) "> 0 | \ (P (x)\) "> 0.2 | \ (X* p (x)\) "> (0) (0.2) = 0 | \ (x -\ mu) ^ {2} P (x)\) "> (0 - 1.1) 2 (0.2) = 0.242 |
| \ (x\) ">1 | \ (P (x)\) "> 0.5 | \ (X* p (x)\) "> (1) (0.5) = 0.5 | \ (x -\ mu) ^ {2} P (x)\) "> (1 - 1.1) 2 (0.5) = 0.005 |
| \ (x\) "> 2 | \ (P (x)\) "> 0.3 | \ (X* p (x)\) "> (2) (0.3) = 0.6 | \ (x -\ mu) ^ {2} P (x)\) "> (2 - 1.1) 2 (0.3) = 0.243 |
Ongeza safu ya mwisho katika meza. \(0.242 + 0.005 + 0.243 = 0.490\). Kupotoka kwa kawaida ni mizizi ya mraba ya 0.49, au\(\sigma = \sqrt{0.49} = 0.7\)
Kwa ujumla kwa mgawanyo uwezekano, tunatumia calculator au kompyuta kuhesabu\(\mu\) na\(\sigma\) kupunguza kosa roundoff. Kwa mgawanyo fulani wa uwezekano, kuna fomu za muda mfupi za kuhesabu\(\mu\) na\(\sigma\).
Mfano\(\PageIndex{5}\)
Toss haki, sita upande mmoja kufa mara mbili. Hebu\(X\) = idadi ya nyuso zinazoonyesha idadi hata. Kujenga meza kama Jedwali na mahesabu ya maana\(\mu\) na kiwango kupotoka\(\sigma\) ya\(X\).
Suluhisho
Tossing moja haki sita upande mmoja kufa mara mbili ina sawa sampuli nafasi kama tossing mbili haki sita upande mmoja dice. Nafasi ya sampuli ina matokeo 36:
| (1, 1) | (1, 2) | (1, 3) | (1, 4) | (1, 5) | (1, 6) |
| (2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) |
| (3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) |
| (4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) |
| (5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) |
| (6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) |
Tumia nafasi ya sampuli ili kukamilisha meza ifuatayo:
| \(x\) | \(P(x)\) | \(xP(x)\) | \((x – \mu)^{2} ⋅ P(x)\) |
|---|---|---|---|
| \ (x\)” style="wima align:katikati; "> 0 | \ (P (x)\)” style="wima align:katikati; ">\(\dfrac{9}{36}\) | \ (XP (x)\)” style="wima align:katikati; "> 0 | \ (x -\ mu) ^ {2} 合 P (x)\)” style="wima align:katikati; ">\((0 – 1)^{2} ⋅ \dfrac{9}{36} = \dfrac{9}{36}\) |
| \ (x\)” style="wima align:katikati; "> 1 | \ (P (x)\)” style="wima align:katikati; ">\(\dfrac{18}{36}\) | \ (XP (x)\)” style="wima align:katikati; ">\(\dfrac{18}{36}\) | \ (x -\ mu) ^ {2} 合 P (x)\)” style="wima align:katikati; ">\((1 – 1)^{2} ⋅ \dfrac{18}{36} = 0\) |
| \ (x\)” style="wima align:katikati; "> 2 | \ (P (x)\)” style="wima align:katikati; ">\(\dfrac{9}{36}\) | \ (XP (x)\)” style="wima align:katikati; ">\(\dfrac{18}{36}\) | \ (x -\ mu) ^ {2} 合 P (x)\)” style="wima align:katikati; ">\((1 – 1)^{2} ⋅ \dfrac{9}{36} = \dfrac{9}{36}\) |
Ongeza maadili katika safu ya tatu ili kupata thamani inayotarajiwa:\(\mu\) =\(\dfrac{36}{36}\) = 1. Tumia thamani hii kukamilisha safu ya nne.
Ongeza maadili katika safu ya nne na kuchukua mizizi ya mraba ya jumla:
\[\sigma = \sqrt{\dfrac{18}{36}} \approx 0.7071.\]
Mfano\(\PageIndex{6}\)
Mnamo Mei 11, 2013 saa 9:30 jioni, uwezekano wa kuwa shughuli za tetemeko la wastani (tetemeko moja la wastani) lingetokea katika masaa 48 ijayo nchini Iran ilikuwa takriban 21.42%. Tuseme unafanya bet kwamba tetemeko la ardhi la wastani litatokea nchini Iran wakati huu. Kama kushinda bet, wewe kushinda $50. Ukipoteza bet, kulipa $20. Hebu X = kiasi cha faida kutoka bet.
\(P(\text{win}) = P(\text{one moderate earthquake will occur}) = 21.42%\)
\(P(\text{loss}) = P(\text{one moderate earthquake will not occur}) = 100% – 21.42%\)
Kama bet mara nyingi, wewe kuja nje mbele? Eleza jibu lako katika sentensi kamili kwa kutumia namba. ni kupotoka kiwango cha\(X\) nini? Jenga meza sawa na Jedwali na Jedwali ili kukusaidia kujibu maswali haya.
Jibu
| \(x\) | \(P(x)\) | \(xP(x)\) | \((x – \mu^{2})P(x)\) | |
|---|---|---|---|---|
| kushinda | \ (x\)” style="wima align:katikati; "> 50 | \ (P (x)\)” style="wima align:katikati; "> 0.2142 | \ (XP (x)\)” style="wima align:katikati; "> 10.71 | \ (x —\ mu^ {2}) P (x)\)” style="wima align:katikati; "> [50 — (—5.006)] 2 (0.2142) = 648.0964 |
| hasara | \ (x\)” style="wima align:katikati; ">—20 | \ (P (x)\)” style="wima align:katikati; "> 0.7858 | \ (XP (x)\)” style="wima align:katikati; "> —15.716 | \ (x —\ mu^ {2}) P (x)\)” style="wima align:katikati; "> [—20 — (—5.006)] 2 (0.7858) = 176.6636 |
Maana = Thamani inayotarajiwa = 10.71 + (—15.716) = —5.006.
Ikiwa unafanya bet hii mara nyingi chini ya hali hiyo, matokeo yako ya muda mrefu yatakuwa hasara ya wastani ya $5.01 kwa bet.
Mkengeuko\(= \sqrt{648.0964+176.6636} \approx 28.7186\)
Zoezi\(\PageIndex{6}\)
Mnamo Mei 11, 2013 saa 9:30 jioni, uwezekano wa kuwa shughuli za tetemeko la wastani (tetemeko moja la wastani) lingetokea katika masaa 48 ijayo nchini Japan ilikuwa karibu 1.08%. Wewe bet kwamba tetemeko la ardhi wastani kutokea katika Japan katika kipindi hiki. Kama kushinda bet, wewe kushinda $100. Ukipoteza bet, kulipa $10. Hebu\(X\) = kiasi cha faida kutoka bet. Kupata maana na kiwango kupotoka ya\(X\).
Jibu
| \(x\) | \(P(x)\) | \(x ⋅ P(x)\) | \((x - \mu^{2}) ⋅ P(x)\) | |
|---|---|---|---|---|
| kushinda | \ (x\)” style="wima align:katikati; "> 100 | \ (P (x)\)” style="wima align:katikati; "> 0.0108 | \ (x) P (x)\)” style="wima align:katikati; "> 1.08 | \ (x -\ mu^ {2}) 合 P (x)\)” style="wima align:katikati; "> [100 — (—8.812)] 2 合 0.0108 = 127.8726 |
| hasara | \ (x\)” style="wima align:katikati; ">—10 | \ (P (x)\)” style="wima align:katikati; "> 0.9892 | \ (x 合 P (x)\)” style="wima align:katikati; "> —9.892 | \ (x -\ mu^ {2}) 合 P (x)\)” style="wima align:katikati; "> [—10 — (—8.812)] 2 合 0.9892 = 1.3961 |
Maana = Thamani inayotarajiwa\(= \mu = 1.08 + (–9.892) = –8.812\)
Ikiwa unafanya bet hii mara nyingi chini ya hali hiyo, matokeo yako ya muda mrefu yatakuwa hasara ya wastani ya $8.81 kwa bet.
Mkengeuko\(= \sqrt{127.7826+1.3961} \approx 11.3696\)
Baadhi ya kazi za kawaida za uwezekano wa kipekee ni binomial, jiometri, hypergeometric, na Poisson. Kozi nyingi za msingi hazifunika kijiometri, hypergeometric, na Poisson. Mwalimu wako atakujulisha kama yeye anataka kufunika mgawanyo huu.
Kazi ya usambazaji uwezekano ni mfano. Unajaribu kukabiliana na tatizo la uwezekano katika muundo au usambazaji ili kufanya mahesabu muhimu. Mgawanyo huu ni zana za kufanya kutatua matatizo ya uwezekano iwe rahisi. Kila usambazaji una sifa zake maalum. Kujifunza sifa kunawezesha kutofautisha kati ya mgawanyo tofauti.
Muhtasari
thamani inatarajiwa, au maana, ya discrete random variable anatabiri matokeo ya muda mrefu ya majaribio ya takwimu ambayo imekuwa mara kwa mara mara nyingi. Kupotoka kwa kiwango cha usambazaji wa uwezekano hutumiwa kupima tofauti ya matokeo iwezekanavyo.
Mapitio ya Mfumo
- Maana au Thamani inayotarajiwa:\(\mu = \sum_{x \in X}xP(x)\)
- Mkengeuko wa kawaida:\(\sigma = \sqrt{\sum_{x \in X}(x - \mu)^{2}P(x)}\)
faharasa
- Thamani inatarajiwa
- inatarajiwa hesabu wastani wakati majaribio ni mara nyingi mara nyingi; pia hujulikana maana. Nukuu:\(\mu\). Kwa discrete random variable (RV) na uwezekano usambazaji kazi\(P(x)\), ufafanuzi pia inaweza kuandikwa katika fomu\(\mu = \sum{xP(x)}\).
- Maana
- idadi ambayo hatua tabia ya kati; jina la kawaida kwa maana ni 'wastani.' Neno 'maana' ni fomu iliyofupishwa ya 'maana ya hesabu. ' Kwa ufafanuzi, maana ya sampuli (iliyopigwa na\(\bar{x}\)) ni\(\bar{x} = \dfrac{\text{Sum of all values in the sample}}{\text{Number of values in the sample}}\) na maana ya idadi ya watu (iliyoonyeshwa na\(\mu\)) ni\(\mu = \dfrac{\text{Sum of all values in the population}}{\text{Number of values in the population}}\).
- Maana ya Usambazaji wa Uwezekano
- wastani wa muda mrefu wa majaribio mengi ya majaribio ya takwimu
- Kupotoka kwa kiwango cha Usambazaji wa Uwezekano
- idadi inayopima jinsi matokeo ya majaribio ya takwimu yanatoka kwa maana ya usambazaji
- Sheria ya Idadi Kubwa
- Kama idadi ya majaribio katika jaribio la uwezekano huongezeka, tofauti kati ya uwezekano wa kinadharia wa tukio na uwezekano wa mzunguko wa jamaa unakaribia sifuri.
Marejeo
- Class Catalogue katika Florida State University Inapatikana mtandaoni kwenye Apps.oti.fsu.edu/Registrarco... ArchFormLegacy (imefikia Mei 15, 2013).
- “Matetemeko ya Dunia: Habari za Tetemeko la ardhi na Mambo muhimu,” Tetemeko la ardhi, 2012. www.world-etearquakes.com/ind... thq_prediction (kupatikana Mei 15, 2013).