<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/591467c2-5f58-42d5-aedf-5c05ade7170e/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls14 ws11"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls14 ws11">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls14 ws11"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 ls14 ws11">[Disciplina - Resumo] </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 ls14 ws11"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 ls14 ws11">ESPERANÇA E VARIÂNCIA DE </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 ls14 ws11">VARIÁVEIS ALEATÓRIAS DISCRETAS </div><div class="t m0 x1 h6 y9 ff3 fs3 fc1 sc0 ls14 ws11">Esperança, valor esperado ou média de uma V<span class="_0 blank"> </span>.A discreta </div><div class="t m0 x1 h7 ya ff2 fs4 fc0 sc0 ls14 ws11">Se X é uma v.a. discreta com função de probabilidade <span class="ff4">\ue74c</span></div><div class="t m0 x3 h8 yb ff4 fs5 fc0 sc0 ls0">\uebd1<span class="fs4 ls14 ws0 v1">(\ue754 )<span class="ff2 ws11"> que assume valores </span></span></div><div class="t m0 x1 h7 yc ff4 fs4 fc0 sc0 ls14">\ue754</div><div class="t m0 x4 h8 yd ff4 fs5 fc0 sc0 ls1">\ueb35<span class="fs4 ls14 ws1 v1">, \ue754</span><span class="ls2">\ueb36<span class="fs4 ls14 ws1 v1">, \u2026 , \ue754<span class="_1 blank"></span><span class="fs5 ls3 v2">\uebe1<span class="ff2 fs4 ls14 ws11 v1">, então a esperança de X é: </span></span></span></span></div><div class="t m0 x5 h9 ye ff4 fs4 fc0 sc0 ls4">\ue727<span class="ls14 ws2 v3">[</span><span class="ls5">\ue73a<span class="ls6 v3">]</span><span class="ls14 ws3">=<span class="_2 blank"> </span>\uedcd \ue754<span class="fs5 ls7 v2">\uebdc</span><span class="ws11"> \ue74c</span></span></span></div><div class="t m0 x6 ha yf ff4 fs5 fc0 sc0 ls0">\uebd1<span class="fs4 ls14 ws2 v4">(<span class="v5">\ue754</span></span><span class="ls7">\uebdc<span class="fs4 ls14 v4">)</span></span></div><div class="t m0 x7 hb y10 ff4 fs5 fc0 sc0 ls14">\uebe1</div><div class="t m0 x8 hc y11 ff4 fs5 fc0 sc0 ls14 ws4">\uebdc \ueb40\ueb35<span class="_3 blank"> </span><span class="fs4 ws11 v6">=<span class="_4 blank"> </span>\uedcd<span class="_5 blank"> </span>\ue754<span class="_0 blank"> </span> \ue74c</span></div><div class="t m0 x9 h8 yf ff4 fs5 fc0 sc0 ls0">\uebd1<span class="fs4 ls14 ws0 v1">(\ue754 )</span></div><div class="t m0 xa hc y11 ff4 fs5 fc0 sc0 ls14 ws11">\uebeb \u2208 \uebc2\uebe0<span class="_0 blank"> </span>(\uebd1)<span class="_6 blank"> </span><span class="fs4 v6">=<span class="_7 blank"> </span>\uedcd<span class="_8 blank"> </span>\ue754<span class="_0 blank"> </span> \ue74c</span></div><div class="t m0 xb h8 yf ff4 fs5 fc0 sc0 ls0">\uebd1<span class="fs4 ls14 ws0 v1">(\ue754 )</span></div><div class="t m0 xc hb y11 ff4 fs5 fc0 sc0 ls14 ws11">\uebeb \u2208 \u211d</div><div class="t m0 xd hd ye ff2 fs4 fc0 sc0 ls14 ws11"> </div><div class="t m0 x1 he y12 ff2 fs4 fc1 sc0 ls14 ws5">Propriedades:<span class="ff3 ws11"> </span></div><div class="t m0 x1 h7 y13 ff2 fs4 fc2 sc0 ls14 ws11">Sejam X e Y v.a\u2019s discretas e <span class="ff4 ws6">\ue887, \ue888<span class="_2 blank"> </span>\u2208<span class="_2 blank"> </span>\u211d</span><span class="ls8">.</span><span class="ff3 fs3 fc1"> </span></div><div class="t m0 xe h9 y14 ff2 fs4 fc0 sc0 ls14 ws5">1.<span class="ff5 ls9 ws11"> <span class="ff4 ls4">\ue727<span class="ls14 ws2 v3">[</span><span class="lsa">\ue73d<span class="lsb v3">]</span><span class="ls14 ws7">=<span class="_2 blank"> </span>\ue73d </span></span></span></span><span class="ws11"> </span></div><div class="t m0 xe h9 y15 ff2 fs4 fc0 sc0 ls14 ws5">2.<span class="ff5 lsc ws11"> <span class="ff4 ls4">\ue727<span class="ls14 ws2 v3">[</span><span class="ls14 ws8">\ue73d<span class="_9 blank"> </span>+<span class="_9 blank"> </span>\ue73e\ue73a<span class="lsb v3">]</span><span class="ws11">=<span class="_2 blank"> </span>\ue73d<span class="_9 blank"> </span>+ \ue73e<span class="_0 blank"> </span> \ue727<span class="_a blank"> </span>[\ue73a<span class="_0 blank"> </span>]<span class="ff2"> </span></span></span></span></span></div><div class="t m0 xe h9 y16 ff2 fs4 fc0 sc0 ls14 ws5">3.<span class="ff5 lsd ws11"> <span class="ff4 ls4">\ue727<span class="ls14 ws2 v3">[</span><span class="ls14 ws9">\ue73a<span class="_9 blank"> </span>+<span class="_9 blank"> </span>\ue73b<span class="ls6 v3">]</span><span class="wsa">=<span class="_2 blank"> </span>\ue727 <span class="ws2 v3">[</span><span class="ls5">\ue73a<span class="lse v3">]</span></span><span class="wsb">+<span class="_9 blank"> </span>\ue727 [\ue73b ]</span></span></span></span></span><span class="ws11"> </span></div><div class="t m0 xe h9 y17 ff2 fs4 fc0 sc0 ls14 ws5">4.<span class="ff5 lsf ws11"> </span><span class="ff4 wsc">min<span class="_b blank"> </span>\ue73a<span class="_c blank"> </span>\u2264<span class="_2 blank"> </span>\ue727 <span class="ws2 v3">[</span><span class="ls5">\ue73a<span class="lsb v3">]</span></span><span class="wsd">\u2264<span class="_2 blank"> </span>max<span class="_b blank"> </span>\ue73a </span></span><span class="ws11"> </span></div><div class="t m0 xe h9 y18 ff2 fs4 fc0 sc0 ls14 ws5">5.<span class="ff5 ls10 ws11"> <span class="ff4 ls4">\ue727<span class="ls14 ws2 v3">[</span><span class="ls11">\ue743<span class="ls14 ws2 v3">(</span><span class="ls5">\ue73a<span class="ls14 wse v3">)] </span><span class="ls14 wsc">\u2260<span class="_2 blank"> </span>\ue743(\ue727 <span class="ws2 v3">[</span></span>\ue73a<span class="ls14 ws2 v3">]<span class="v5">)</span></span></span></span></span></span><span class="ws11">, e<span class="_0 blank"> </span>xceto se <span class="ff4 ls11">\ue743</span> for função linear. </span></div><div class="t m0 xf hd y19 ff2 fs4 fc0 sc0 ls14 ws11"> </div><div class="t m0 x1 h6 y1a ff3 fs3 fc1 sc0 ls14 ws11">Variância de uma v.a discreta </div><div class="t m0 x1 h7 y1b ff2 fs4 fc0 sc0 ls14 ws11">Seja X uma v.a discreta com função de probabilidade <span class="ff4">\ue74c</span></div><div class="t m0 x10 h8 y1c ff4 fs5 fc0 sc0 ls0">\uebd1<span class="fs4 ls14 ws0 v1">(\ue754 )<span class="ff2 ws11">. A variância de X é </span></span></div><div class="t m0 x1 hd y1d ff2 fs4 fc0 sc0 ls14 ws11">definida da seguinte maneira: </div><div class="t m0 x11 h9 y1e ff4 fs4 fc0 sc0 ls14 ws11">\ue738\ue73d\ue74e<span class="_0 blank"> </span> <span class="ws2 v3">(</span><span class="ls5">\ue73a<span class="lsb v3">)</span></span><span class="wsc">=<span class="_2 blank"> </span>\ue727 [<span class="ws2 v3">(</span>\ue73a<span class="_9 blank"> </span>\u2212<span class="_9 blank"> </span>\ue727 <span class="ws2 v3">[</span><span class="ls5">\ue73a</span><span class="ws2 v3">])</span><span class="fs5 ls1 v7">\ueb36</span><span class="ws2">]</span></span><span class="ff2"> </span></div><div class="t m0 x12 h9 y1f ff4 fs4 fc0 sc0 ls14 ws11">\ue738\ue73d\ue74e<span class="_0 blank"> </span> <span class="ws2 v3">(</span><span class="ls5">\ue73a<span class="lsb v3">)</span></span><span class="wsf">=<span class="_4 blank"> </span>\uedcd <span class="ws2 v3">(</span><span class="wsa">\ue754<span class="_9 blank"> </span>\u2212<span class="_9 blank"> </span>\ue727 <span class="ws2 v3">[</span><span class="ls5">\ue73a</span><span class="ws2 v3">])</span><span class="fs5 ls2 v7">\ueb36</span></span></span> \ue74c</div><div class="t m0 x13 h8 y20 ff4 fs5 fc0 sc0 ls0">\uebd1<span class="fs4 ls14 ws0 v1">(\ue754 )</span></div><div class="t m0 x14 hb y21 ff4 fs5 fc0 sc0 ls14 ws11">\uebeb \u2208 \uebc2\uebe0<span class="_0 blank"> </span>(\uebd1)</div><div class="t m0 x15 hd y1f ff2 fs4 fc0 sc0 ls14 ws11"> </div><div class="t m0 x1 hd y22 ff2 fs4 fc1 sc0 ls14 ws11">Resultado: </div><div class="t m0 x16 h9 y23 ff4 fs4 fc2 sc0 ls14 ws11">\ue738\ue73d\ue74e<span class="_0 blank"> </span> <span class="ws2 v3">(</span><span class="ls5">\ue73a<span class="ls6 v3">)</span></span>=<span class="_2 blank"> </span> \ue727<span class="_0 blank"> </span><span class="ws2 v3">[</span><span class="ls12">\ue73a<span class="fs5 ls2 v7">\ueb36</span><span class="lse v3">]</span><span class="ls13">\u2212</span></span><span class="ws2 v3">(</span><span class="ls4">\ue727</span><span class="ws2 v3">[</span><span class="ls5">\ue73a</span><span class="ws2 v3">])</span><span class="fs5 ls2 v7">\ueb36</span><span class="ff2"> , onde<span class="ff3"> </span></span></div><div class="t m0 x17 h9 y24 ff4 fs4 fc2 sc0 ls4">\ue727<span class="ls14 ws2 v3">[</span><span class="ls12">\ue73a<span class="fs5 ls2 v7">\ueb36</span><span class="ls6 v3">]</span><span class="ls14 ws10">=<span class="_4 blank"> </span>\uedcd<span class="_5 blank"> </span>\ue754 <span class="fs5 ls1 v7">\ueb36</span><span class="ws11"> \ue74c</span></span></span></div><div class="t m0 x18 h8 y25 ff4 fs5 fc2 sc0 ls0">\uebd1<span class="fs4 ls14 ws0 v1">(\ue754 )</span></div><div class="t m0 xa hb y26 ff4 fs5 fc2 sc0 ls14 ws11">\uebeb \u2208 \uebc2\uebe0<span class="_0 blank"> </span>(\uebd1)</div><div class="t m0 x19 he y24 ff3 fs4 fc2 sc0 ls14 ws11"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/591467c2-5f58-42d5-aedf-5c05ade7170e/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls14 ws11"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls14 ws11">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls14 ws11"> </div></div><div class="t m0 x1 hd y27 ff2 fs4 fc2 sc0 ls14 ws11"> <span class="fc1">Propriedades: </span> </div><div class="t m0 x1 h7 y28 ff2 fs4 fc2 sc0 ls14 ws11"> Sejam X e Y v.a\u2019s quaisquer e <span class="ff4 ws7">\ue73d ,<span class="_b blank"> </span>\ue73e<span class="_c blank"> </span>\u2208<span class="_2 blank"> </span>\u211d</span><span class="ws5">.</span><span class="ff3"> </span></div><div class="t m0 xe h9 y29 ff3 fs4 fc2 sc0 ls14 ws12">1.<span class="ff6 ls15 ws11"> </span><span class="ff4 ws13">\ue738\ue73d\ue74e<span class="ws2 v3">(</span><span class="lsa">\ue73d<span class="lsb v3">)</span></span><span class="ws14">= 0<span class="_0 blank"> </span></span></span><span class="ws11"> </span></div><div class="t m0 xe h9 y2a ff3 fs4 fc2 sc0 ls14 ws12">2.<span class="ff6 ls16 ws11"> <span class="ff4 ls14">\ue738\ue73d\ue74e<span class="_0 blank"> </span> <span class="ws2 v3">(</span><span class="ws8">\ue73d<span class="_9 blank"> </span>+<span class="_9 blank"> </span>\ue73e\ue73a<span class="ls6 v3">)</span><span class="ws15">=<span class="_2 blank"> </span>\ue73e <span class="fs5 ls2 v7">\ueb36</span></span></span> \ue738\ue73d\ue74e<span class="_0 blank"> </span> (\ue73a<span class="_0 blank"> </span>)<span class="ff3"> </span></span></span></div><div class="t m0 xe h9 y2b ff3 fs4 fc2 sc0 ls14 ws12">3.<span class="ff6 lsf ws11"> </span><span class="ff4 ws13">\ue738\ue73d\ue74e <span class="ws2 v3">(</span><span class="ls5">\ue73a<span class="ls6 v3">)</span></span><span class="ws16">\u2265 0</span></span><span class="ws11"> </span></div><div class="t m0 xe h9 y2c ff3 fs4 fc2 sc0 ls14 ws12">4.<span class="ff6 ls17 ws11"> <span class="ff2 ls14">Em geral, <span class="ff4">\ue738\ue73d\ue74e<span class="_0 blank"> </span> <span class="ws2 v3">(</span><span class="ws9">\ue73a<span class="_9 blank"> </span>+<span class="_9 blank"> </span>\ue73b<span class="ls6 v3">)</span></span>\u2260<span class="_2 blank"> </span>\ue738\ue73d\ue74e<span class="_0 blank"> </span> <span class="ws2 v3">(</span><span class="ls5">\ue73a<span class="lse v3">)</span></span>+ \ue738\ue73d\ue74e<span class="_0 blank"> </span> (\ue73b<span class="_0 blank"> </span>)<span class="ff3"> </span></span></span></span></div><div class="t m0 x1 he y2d ff3 fs4 fc2 sc0 ls14 ws11"> </div><div class="t m0 x1 h6 y2e ff3 fs3 fc1 sc0 ls14 ws11">Desvio padrão de uma v.a </div><div class="t m0 x1 hd y2f ff2 fs4 fc0 sc0 ls14 ws11">Seja X uma v.a. O desvio padrão de X é dado por: </div><div class="t m0 x1a h9 y30 ff4 fs4 fc0 sc0 ls14 ws11">\ue726\ue732<span class="_0 blank"> </span> <span class="ws2 v3">(</span><span class="ls5">\ue73a<span class="ls6 v3">)</span><span class="ls18">=<span class="ls19 v0">\ueda5</span></span></span>\ue738\ue73d\ue74e<span class="_0 blank"> </span> <span class="ws2 v3">(</span><span class="ls5">\ue73a</span><span class="ws2 v3">)</span><span class="ff2"> </span></div><div class="t m0 x1 hd y31 ff2 fs4 fc1 sc0 ls14 ws11">Propriedades: </div><div class="t m0 xe h9 y32 ff2 fs4 fc0 sc0 ls14 ws5">1.<span class="ff5 ls9 ws11"> <span class="ff4 ls14">DP <span class="ws2 v3">(</span><span class="ls19">a<span class="ls6 v3">)</span></span><span class="ws14">= 0</span><span class="ff2"> </span></span></span></div><div class="t m0 xe h9 y33 ff2 fs4 fc0 sc0 ls14 ws5">2.<span class="ff5 lsc ws11"> <span class="ff4 ls14">\ue726\ue732<span class="_0 blank"> </span> <span class="ws2 v3">(</span><span class="ws8">\ue73d<span class="_9 blank"> </span>+<span class="_9 blank"> </span>\ue73e\ue73a<span class="ls6 v3">)</span><span class="ls18">=</span><span class="ws2 v3">|</span><span class="ls1a">\ue73e<span class="ls1b v3">|</span></span></span>\ue726\ue732<span class="_0 blank"> </span> (\ue73a<span class="_0 blank"> </span>)<span class="ff2"> </span></span></span></div><div class="t m0 xe h9 y34 ff2 fs4 fc0 sc0 ls14 ws5">3.<span class="ff5 lsd ws11"> </span><span class="ff4 wsd">\ue726\ue732 <span class="ws2 v3">(</span><span class="ls5">\ue73a<span class="lsb v3">)</span></span><span class="ws17">\u2265 0</span></span><span class="ws11"> </span></div><div class="t m0 x1 hd y35 ff2 fs4 fc0 sc0 ls14 ws11"> </div><div class="t m0 x1 hf y36 ff2 fs0 fc0 sc0 ls14 ws11"> </div><div class="t m0 x1 hf y37 ff2 fs0 fc0 sc0 ls14 ws11"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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