8.5-bo'lim : Ehtimollik

1. Mayli, \[f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{3}{4}\left( {2x - {x^2}} \right)}&{{\mbox{if }}0 \le x \le 2}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
   1. \(f\left( x \right)\) ehtimollik zichligi funksiyasi ekanligini ko'rsating.
   2. \(P\left( {X \le 0.25} \right)\) ni toping.
   3. \(P\left( {X \ge 1.4} \right)\) ni toping.
   4. \(P\left( {0.1 \le X \le 1.2} \right)\) ni toping.
   5. \(X\) ning o'rtacha qiymatini aniqlang.
2. Mayli, \[f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{4}{{\ln \left( 3 \right)\left( {4x + {x^2}} \right)}}}&{{\mbox{if 1}} \le x \le 6}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
   1. \(f\left( x \right)\) ehtimollik zichligi funksiyasi ekanligini ko'rsating.
   2. \(P\left( {X \le 1} \right)\) ni toping.
   3. \(P\left( {X \ge 5} \right)\) ni toping.
   4. \(P\left( {1 \le X \le 5} \right)\) ni toping.
   5. \(X\) ning o'rtacha qiymatini aniqlang.
3. Mayli, \[f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{1}{{10}}\left( {1 + \sin \left( {\pi x - \frac{\pi }{2}} \right)} \right)}&{{\mbox{if 0}} \le x \le 10}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
   1. \(f\left( x \right)\) ehtimollik zichligi funksiyasi ekanligini ko'rsating.
   2. \(P\left( {X \le 3} \right)\) ni toping.
   3. \(P\left( {X \ge 5} \right)\) ni toping.
   4. \(P\left( {2.5 \le X \le 7} \right)\) ni toping.
   5. \(X\) ning o'rtacha qiymatini aniqlang.
4. Batareyaning ishlash muddatining ehtimollik zichligi funksiyasi quyidagi funktsiya bilan berilgan, bu erda t yillarda. \[f\left( t \right) = \left\{ {\begin{array}{ll}{1.25{{\bf{e}}^{ - \,1.25t}}}&{{\mbox{if }}t \ge 0}\\0&{{\mbox{if }}t < 0}\end{array}} \right.\]
   1. \(f\left( t \right)\) ehtimollik zichligi funksiyasi ekanligini tekshiring.
   2. Batareyaning ishlash muddati 10 oydan kam bo'lishi ehtimoli qanday?
   3. Batareyaning 2 yildan ortiq xizmat qilish ehtimoli qanday?
   4. Batareyaning 1,5 dan 4 yilgacha xizmat qilish ehtimoli qanday?
   5. Batareyalarning ishlash muddatining o'rtacha qiymatini aniqlang.
5. Ba'zi tajribaning muvaffaqiyatli natijasining ehtimollik zichligi funktsiyasi quyidagi funktsiya bilan berilgan, bu erda t daqiqalarda. \[f\left( t \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{1}{9}\,t\,{{\bf{e}}^{ - \,\,\frac{t}{3}}}}&{{\mbox{if }}t \ge 0}\\0&{{\mbox{if }}t < 0}\end{array}} \right.\]
   1. \(f\left( t \right)\) ehtimollik zichligi funksiyasi ekanligini tekshiring.
   2. Muvaffaqiyatli natija 12 daqiqadan kamroq vaqt ichida sodir bo'lish ehtimoli qanday?
   3. 25 daqiqadan so'ng muvaffaqiyatli natijaga erishish ehtimoli qanday?
   4. Muvaffaqiyatli natijaning 10 dan 75 daqiqagacha bo'lish ehtimoli qanday?
   5. Tajribadan muvaffaqiyatli natijaning o'rtacha vaqti qancha?
6. \(c\) qiymatini aniqlang, buning uchun quyidagi funktsiya ehtimollik zichligi funktsiyasi bo'ladi. \[f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{c\left( {12{x^4} - {x^5}} \right)}&{{\mbox{if }}0 \le x \le 12}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
7. Ushbu muammo uchun quyidagi funktsiyadan foydalaning va \(a > 0\) ni qabul qiling. \[f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{c\,{{\bf{e}}^{ - \,\,\frac{1}{a}\,\,x}}}&{x \ge 0}\\0&{x < 0}\end{array}} \right.\]
   1. \(c\) qiymatini aniqlang, bu funksiya uchun ehtimollik zichligi funksiyasi bo'ladi.
   2. Birinchi qismda topilgan \(c\) qiymatidan foydalanib, ehtimollik zichligi funktsiyasining o'rtacha qiymatini aniqlang.