https://peda.net/id/a70c083231e 2020-01-08T08:32:33+02:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/c11743c2536 2020-02-20T00:47:06+02:00 <div><br/> Tiheysfunktion määritelmä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%5Cge0" alt="f\left(x\right)\ge0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cint_%7B-%5Cinfty%7D%5E%7B%5C%20%5Cinfty%7Df%5Cleft(x%5Cright)dx%3D1" alt="A=\int_{-\infty}^{\ \infty}f\left(x\right)dx=1"/></div> <div>Koska f(x) on oltava suurempi tai yhtäsuuri kuin 0, tarkastellaan ylempi funktion a(x+3) integraali funktio</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-2%7D%5E2a%5Cleft(x%2B3%5Cright)%3D1" alt="\int_{-2}^2a\left(x+3\right)=1"/></div> <div>Koska</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-2%7D%5E2%5Cleft(x%2B3%5Cright)%3D12" alt="\int_{-2}^2\left(x+3\right)=12"/></div> <div>a on oltava <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B12%7D" alt="\frac{1}{12}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cint_%7B-2%7D%5E2%5Cfrac%7B1%7D%7B12%7D%5Cleft(x%2B3%5Cright)%3D1" alt="\int_{-2}^2\frac{1}{12}\left(x+3\right)=1"/></div> <div>eli</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%5Cbegin%7Bcases%7D%0A%5Cfrac%7B1%7D%7B12%7D%5Cleft(x%2B3%5Cright)%7B%2C%7D%26kun%5C%20-2%5Cle%20x%5Cle2%5C%5C%0A0%7B%2C%7D%26kun%5C%20x%3C-2%5C%20tai%5C%20x%3E2%0A%5Cend%7Bcases%7D" alt="f\left(x\right)\begin{cases}&#10;\frac{1}{12}\left(x+3\right){,}&amp;kun\ -2\le x\le2\\&#10;0{,}&amp;kun\ x&lt;-2\ tai\ x&gt;2&#10;\end{cases}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0Ax%26P%5Cleft(X%3Dx%5Cright)%5C%5C%0A%5Chline%0A1%26%5Cfrac%7B1%7D%7B12%7D%5Cleft(1%2B3%5Cright)%3D%5Cfrac%7B1%7D%7B3%7D%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;x&amp;P\left(X=x\right)\\&#10;\hline&#10;1&amp;\frac{1}{12}\left(1+3\right)=\frac{1}{3}&#10;\end{array}"/></div> 2020-02-20T00:47:06+02:00 https://peda.net/id/7e1b2a2446d 2020-02-04T00:43:39+02:00 <span>Tutkimuksessa todettiin, että 200 gramman keksipakkausten massan keskiarvo oli 204 g ja keskihajonta 6 g. Oletetaan, että massa on normaalisti jakautunut. Kuinka monella prosentilla pakkauksista massa oli alle 200 g? Kuinka monella prosentilla pakkauksista massa oli välillä 200 g - 210 g? (K01/7)</span><br/> <div> </div> <div>Ratkaisut:</div> <div>Satunnaismuuttuja X noudattaa normaalijakaumaa<img src="https://math-demo.abitti.fi/math.svg?latex=N%5Cleft(204%7B%2C%7D6%5Cright)" alt="N\left(204{,}6\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%3C200%5Cright)%3DP%5Cleft(X%5Cle200%5Cright)" alt="P\left(X&lt;200\right)=P\left(X\le200\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=Z%3D%5Cfrac%7Bx-204%7D%7B6%7D" alt="Z=\frac{x-204}{6}"/><span>noudattaa normitettua normaalijakaumaa.</span><img src="https://math-demo.abitti.fi/math.svg?latex=N%5Cleft(0%7B%2C%7D1%5Cright)" alt="N\left(0{,}1\right)"/> <div>Siis</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cle200%5Cright)%3DP%5Cleft(Z%5Cle%5Cfrac%7B200-204%7D%7B6%7D%5Cright)%3DP%5Cleft(Z%5Cle-0%7B%2C%7D667%5Cright)%3D%5CPhi%5Cleft(0%7B%2C%7D667%5Cright)%3D1-%5CPhi%5Cleft(0%7B%2C%7D667%5Cright)%3D1-0%7B%2C%7D7486%3D0%7B%2C%7D2514%3D14%5C%25" alt="P\left(X\le200\right)=P\left(Z\le\frac{200-204}{6}\right)=P\left(Z\le-0{,}667\right)=\Phi\left(0{,}667\right)=1-\Phi\left(0{,}667\right)=1-0{,}7486=0{,}2514=14\%"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(200%5Cle%20X%5Cle210%5Cright)%3DP%5Cleft(%5Cfrac%7B200-204%7D%7B6%7D%5Cle%20Z%5Cle%5Cfrac%7B210-204%7D%7B6%7D%5Cright)%3DP%5Cleft(-0%7B%2C%7D67%5Cle%20Z%5Cle1%5Cright)%3D%5CPhi%5Cleft(1%5Cright)-%5CPhi%5Cleft(-0%7B%2C%7D67%5Cright)%3D%5CPhi%5Cleft(1%5Cright)-1%2B%5CPhi%5Cleft(0%7B%2C%7D67%5Cright)%3D0%7B%2C%7D8413-1%2B0%7B%2C%7D7486%3D0%7B%2C%7D588852%5Capprox0%7B%2C%7D59%3D0%7B%2C%7D59%5C%25" alt="P\left(200\le X\le210\right)=P\left(\frac{200-204}{6}\le Z\le\frac{210-204}{6}\right)=P\left(-0{,}67\le Z\le1\right)=\Phi\left(1\right)-\Phi\left(-0{,}67\right)=\Phi\left(1\right)-1+\Phi\left(0{,}67\right)=0{,}8413-1+0{,}7486=0{,}588852\approx0{,}59=0{,}59\%"/><br/> <div> </div> <div> </div> <div>Oletetaan, että väestön älykkyysosamäärä noudattaa normaalijakaumaa N(100,15). Määritä odotusarvon ympäriltä symmetrinen väli, johon kuuluu täsmälleen puolet väestöstä. (K15/6) </div> <div>Älykkyysosamäärä satunnaismuuttuja X</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(100-a%5Cle%20X%5Cle100%2Ba%5Cright)%3D0%7B%2C%7D50" alt="P\left(100-a\le X\le100+a\right)=0{,}50"/></div> <span>Normitetaan satunnaismuuttuja X noudattaamaan jakauma N(0,1). Jolloin</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(%5Cfrac%7B100-a-100%7D%7B15%7D%5Cle%20Z%5Cle%5Cfrac%7B100%2Ba-100%7D%7B15%7D%5Cright)%3DP%5Cleft(-%5Cfrac%7Ba%7D%7B15%7D%5Cle%20Z%5Cle%5Cfrac%7Ba%7D%7B15%7D%5Cright)%3D0%7B%2C%7D50" alt="P\left(\frac{100-a-100}{15}\le Z\le\frac{100+a-100}{15}\right)=P\left(-\frac{a}{15}\le Z\le\frac{a}{15}\right)=0{,}50"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5CPhi%5Cleft(%5Cfrac%7Ba%7D%7B15%7D%5Cright)-%5CPhi%5Cleft(-%5Cfrac%7B1%7D%7B15%7D%5Cright)%3D0%7B%2C%7D50" alt="\Phi\left(\frac{a}{15}\right)-\Phi\left(-\frac{1}{15}\right)=0{,}50"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5CPhi%5Cleft(%5Cfrac%7Ba%7D%7B15%7D%5Cright)-1%2B%5CPhi%5Cleft(-%5Cfrac%7Ba%7D%7B15%7D%5Cright)%3D0%7B%2C%7D50" alt="\Phi\left(\frac{a}{15}\right)-1+\Phi\left(-\frac{a}{15}\right)=0{,}50"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5CPhi%5Cleft(%5Cfrac%7Ba%7D%7B15%7D%5Cright)-1%3D0%7B%2C%7D50" alt="2\Phi\left(\frac{a}{15}\right)-1=0{,}50"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5CPhi%5Cleft(%5Cfrac%7Ba%7D%7B15%7D%5Cright)%3D0%7B%2C%7D75%3D%5CPhi%5Cleft(0%7B%2C%7D67%5Cright)%3D0%7B%2C%7D7486" alt="\Phi\left(\frac{a}{15}\right)=0{,}75=\Phi\left(0{,}67\right)=0{,}7486"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Ba%7D%7B15%7D%5Capprox0%7B%2C%7D67" alt="\frac{a}{15}\approx0{,}67"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Capprox10%7B%2C%7D05" alt="a\approx10{,}05"/><br/> <div>N. 50% väestöstä kuuluu siis älykkyysosamäärävälille [90,100]</div> <div> </div> <div>Laskimella </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(100-a%5Cle%20X%5Cle100%2Ba%5Cright)%3DP%5Cleft(X%5Cle100%2Ba%5Cright)-P%5Cleft(X%5Cge100-a%5Cright)" alt="P\left(100-a\le X\le100+a\right)=P\left(X\le100+a\right)-P\left(X\ge100-a\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(X%5Cle100%2Ba%5Cright)-1%2BP%5Cleft(X%5Cle100%2Ba%5Cright)" alt="=P\left(X\le100+a\right)-1+P\left(X\le100+a\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2P%5Cleft(X%5Cle100%2Ba%5Cright)-1%3D0%7B%2C%7D50" alt="2P\left(X\le100+a\right)-1=0{,}50"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(X%5Cle100%2Ba%5Cright)%3D0%7B%2C%7D75" alt="P\left(X\le100+a\right)=0{,}75"/><br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4%7D%7B3-2x%7D%3C0%7B%2C%7D%5C%20x%5Cne%5Cfrac%7B3%7D%7B2%7D" alt="\frac{4}{3-2x}&lt;0{,}\ x\ne\frac{3}{2}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4%7D%7B%5Cfrac%7B3-2x%7D%7B4%7D%7D%3C%5Cfrac%7B0%7D%7B4%7D" alt="\frac{4}{\frac{3-2x}{4}}&lt;\frac{0}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B3-2x%7D%3C0" alt="\frac{1}{3-2x}&lt;0"/></div> <div>Selvitetään nimittäjän merkki</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3-2x%3C0" alt="3-2x&lt;0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-2x%3C-3" alt="-2x&lt;-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E%5Cfrac%7B3%7D%7B2%7D" alt="x&gt;\frac{3}{2}"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"--><br/> <br/> Laatikossa on 7 punaista , 8 sinistä ja 5 mustaa palloa. Millä todennäköisyydellä <div>a) Nostetaan 2 samanvääristä palloa</div> <div>b) Nostetaan vähintään 3 punaista, kun kaikkiaan nostetaan 4 palloa?</div> <div> </div> <div>a) </div> <div>P(2 samanväristä)=P(2p tai 2s tai 2m)</div> <div>(Tapahtumat erillisiä)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(2p%5Cright)%2BP%5Cleft(2s%5Cright)%2BP%5Cleft(2m%5Cright)" alt="=P\left(2p\right)+P\left(2s\right)+P\left(2m\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(1.p%5C%20ja%5C%202.p%5Cright)%2BP%5Cleft(1.s%5C%20ja%5C%202.s%5Cright)%2B%5C%20P%5Cleft(1.m%5C%20ja%5C%202.m%5Cright)" alt="=P\left(1.p\ ja\ 2.p\right)+P\left(1.s\ ja\ 2.s\right)+\ P\left(1.m\ ja\ 2.m\right)"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(1.p%5Cright)%5Ccdot%20P%5Cleft(2.p%5Cright)%2BP%5Cleft(1.s%5Cright)%5Ccdot%20P%5Cleft(2.s%5Cright)%2B%5C%20P%5Cleft(1.m%5Cright)%5Ccdot%20P%5Cleft(2.m%5Cright)" alt="=P\left(1.p\right)\cdot P\left(2.p\right)+P\left(1.s\right)\cdot P\left(2.s\right)+\ P\left(1.m\right)\cdot P\left(2.m\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3DP%5Cleft(1.p%5Cright)%5Ccdot%20P%5Cleft(%5Cfrac%7B2.p%7D%7B1.p%7D%5Cright)%2BP%5Cleft(1.s%5Cright)%5Ccdot%20P%5Cleft(%5Cfrac%7B2.s%7D%7B1.s%7D%5Cright)%2B%5C%20P%5Cleft(1.m%5Cright)%5Ccdot%20P%5Cleft(%5Cfrac%7B2.m%7D%7B1.m%7D%5Cright)" alt="=P\left(1.p\right)\cdot P\left(\frac{2.p}{1.p}\right)+P\left(1.s\right)\cdot P\left(\frac{2.s}{1.s}\right)+\ P\left(1.m\right)\cdot P\left(\frac{2.m}{1.m}\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B7%7D%7B20%7D%5Ccdot%5Cfrac%7B6%7D%7B19%7D%2B%5Cfrac%7B8%7D%7B20%7D%5Ccdot%5Cfrac%7B7%7D%7B19%7D%2B%5Cfrac%7B5%7D%7B20%7D%5Ccdot%5Cfrac%7B4%7D%7B19%7D" alt="=\frac{7}{20}\cdot\frac{6}{19}+\frac{8}{20}\cdot\frac{7}{19}+\frac{5}{20}\cdot\frac{4}{19}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B59%7D%7B190%7D%5Capprox0%7B%2C%7D31" alt="=\frac{59}{190}\approx0{,}31"/></div> <div> </div> <div>b)</div> <div>Kakki alkeistapaukset ovat kaikki mahdolliset 4 pallon joukot eli <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B20%7D%7B4%7D" alt="\binom{20}{4}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(v%C3%A4h%5C%203p%5Cright)%3DP%5Cleft(3p%5C%20tai%5C%204p%5Cright)%3DP%5Cleft(3p%5Cright)%2BP%5Cleft(4p%5Cright)" alt="P\left(väh\ 3p\right)=P\left(3p\ tai\ 4p\right)=P\left(3p\right)+P\left(4p\right)"/></div> Kun 3 punaista nostetaan, nostetaan 1 muuvärinen , eri tapoja on tällöin <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B7%7D%7B3%7D%5Ccdot%5Cbinom%7B13%7D%7B1%7D" alt="\binom{7}{3}\cdot\binom{13}{1}"/> <div>Kun 4 punaista voidaan nostaa <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbinom%7B7%7D%7B4%7D" alt="\binom{7}{4}"/>eri tavalla.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%5Cleft(3p%5Cright)%2BP%5Cleft(4p%5Cright)%3D%5Cfrac%7B%5Cbinom%7B7%7D%7B3%7D%5Ccdot%5Cbinom%7B17%7D%7B1%7D%7D%7B%5Cbinom%7B20%7D%7B4%7D%7D%2B%5Cfrac%7B%5Cbinom%7B7%7D%7B4%7D%7D%7B%5Cbinom%7B20%7D%7B4%7D%7D%3D%5Cfrac%7B98%7D%7B969%7D%5Capprox0%7B%2C%7D10" alt="P\left(3p\right)+P\left(4p\right)=\frac{\binom{7}{3}\cdot\binom{17}{1}}{\binom{20}{4}}+\frac{\binom{7}{4}}{\binom{20}{4}}=\frac{98}{969}\approx0{,}10"/><br/> <div> <div> </div> </div> </div> </div> 2020-02-04T00:42:44+02:00