\( A = \{3, 9, 16, 23, 26\}\) und \( B = \{3, 7, 9, 23, 26\}\).

Bestimmen Sie die folgenden Wahrscheinlichkeiten:

Verteilungsfunktion \( \Phi(x) \) der \(N(0,1)\) Verteilung \[ \begin{array}{l|llllllllll}x & ,00 & ,01 & ,02 & ,03 & ,04& ,05& ,06& ,07& ,08& ,09 \\ \hline0& 0.5& 0.50399& 0.50798& 0.51197& 0.51595& 0.51994& 0.52392& 0.5279& 0.53188& 0.53586\\0.1& 0.53983& 0.5438& 0.54776& 0.55172& 0.55567& 0.55962& 0.56356& 0.56749& 0.57142& 0.57535\\0.2& 0.57926& 0.58317& 0.58706& 0.59095& 0.59483& 0.59871& 0.60257& 0.60642& 0.61026& 0.61409\\0.3& 0.61791& 0.62172& 0.62552& 0.6293& 0.63307& 0.63683& 0.64058& 0.64431& 0.64803& 0.65173\\0.4& 0.65542& 0.6591& 0.66276& 0.6664& 0.67003& 0.67364& 0.67724& 0.68082& 0.68439& 0.68793\\\hline0.5& 0.69146& 0.69497& 0.69847& 0.70194& 0.7054& 0.70884& 0.71226& 0.71566& 0.71904& 0.7224\\0.6& 0.72575& 0.72907& 0.73237& 0.73565& 0.73891& 0.74215& 0.74537& 0.74857& 0.75175& 0.7549\\0.7& 0.75804& 0.76115& 0.76424& 0.7673& 0.77035& 0.77337& 0.77637& 0.77935& 0.7823& 0.78524\\0.8& 0.78814& 0.79103& 0.79389& 0.79673& 0.79955& 0.80234& 0.80511& 0.80785& 0.81057& 0.81327\\0.9& 0.81594& 0.81859& 0.82121& 0.82381& 0.82639& 0.82894& 0.83147& 0.83398& 0.83646& 0.83891\\\hline1& 0.84134& 0.84375& 0.84614& 0.84849& 0.85083& 0.85314& 0.85543& 0.85769& 0.85993& 0.86214\\1.1& 0.86433& 0.8665& 0.86864& 0.87076& 0.87286& 0.87493& 0.87698& 0.879& 0.881& 0.88298\\1.2& 0.88493& 0.88686& 0.88877& 0.89065& 0.89251& 0.89435& 0.89617& 0.89796& 0.89973& 0.90147\\1.3& 0.9032& 0.9049& 0.90658& 0.90824& 0.90988& 0.91149& 0.91309& 0.91466& 0.91621& 0.91774\\1.4& 0.91924& 0.92073& 0.9222& 0.92364& 0.92507& 0.92647& 0.92785& 0.92922& 0.93056& 0.93189\\\hline1.5& 0.93319& 0.93448& 0.93574& 0.93699& 0.93822& 0.93943& 0.94062& 0.94179& 0.94295& 0.94408\\\end{array} \]Sei \(X\) eine \( N(\mu, \sigma=1)\)-verteilte Zufallsvariable für die \( P(X < 0.5) = 0.697\) gilt.

Wie groß ist \( \mu \)?

Verteilungsfunktion \( \Phi(x) \) der \(N(0,1)\) Verteilung \[ \begin{array}{l|llllllllll}x & ,00 & ,01 & ,02 & ,03 & ,04& ,05& ,06& ,07& ,08& ,09 \\ \hline0& 0.5& 0.50399& 0.50798& 0.51197& 0.51595& 0.51994& 0.52392& 0.5279& 0.53188& 0.53586\\0.1& 0.53983& 0.5438& 0.54776& 0.55172& 0.55567& 0.55962& 0.56356& 0.56749& 0.57142& 0.57535\\0.2& 0.57926& 0.58317& 0.58706& 0.59095& 0.59483& 0.59871& 0.60257& 0.60642& 0.61026& 0.61409\\0.3& 0.61791& 0.62172& 0.62552& 0.6293& 0.63307& 0.63683& 0.64058& 0.64431& 0.64803& 0.65173\\0.4& 0.65542& 0.6591& 0.66276& 0.6664& 0.67003& 0.67364& 0.67724& 0.68082& 0.68439& 0.68793\\\hline0.5& 0.69146& 0.69497& 0.69847& 0.70194& 0.7054& 0.70884& 0.71226& 0.71566& 0.71904& 0.7224\\0.6& 0.72575& 0.72907& 0.73237& 0.73565& 0.73891& 0.74215& 0.74537& 0.74857& 0.75175& 0.7549\\0.7& 0.75804& 0.76115& 0.76424& 0.7673& 0.77035& 0.77337& 0.77637& 0.77935& 0.7823& 0.78524\\0.8& 0.78814& 0.79103& 0.79389& 0.79673& 0.79955& 0.80234& 0.80511& 0.80785& 0.81057& 0.81327\\0.9& 0.81594& 0.81859& 0.82121& 0.82381& 0.82639& 0.82894& 0.83147& 0.83398& 0.83646& 0.83891\\\hline1& 0.84134& 0.84375& 0.84614& 0.84849& 0.85083& 0.85314& 0.85543& 0.85769& 0.85993& 0.86214\\1.1& 0.86433& 0.8665& 0.86864& 0.87076& 0.87286& 0.87493& 0.87698& 0.879& 0.881& 0.88298\\1.2& 0.88493& 0.88686& 0.88877& 0.89065& 0.89251& 0.89435& 0.89617& 0.89796& 0.89973& 0.90147\\1.3& 0.9032& 0.9049& 0.90658& 0.90824& 0.90988& 0.91149& 0.91309& 0.91466& 0.91621& 0.91774\\1.4& 0.91924& 0.92073& 0.9222& 0.92364& 0.92507& 0.92647& 0.92785& 0.92922& 0.93056& 0.93189\\\hline1.5& 0.93319& 0.93448& 0.93574& 0.93699& 0.93822& 0.93943& 0.94062& 0.94179& 0.94295& 0.94408\\\end{array} \]Sei \(X\) eine \( N(\mu = 1.105, \sigma=1.179)\)-verteilte Zufallsvariable.

Wie groß ist die Wahrscheinlichkeit \[ P(-0.04 < X ) ?\]

Verteilungsfunktion \( \Phi(x) \) der \(N(0,1)\) Verteilung \[ \begin{array}{l|llllllllll}x & ,00 & ,01 & ,02 & ,03 & ,04& ,05& ,06& ,07& ,08& ,09 \\ \hline0& 0.5& 0.50399& 0.50798& 0.51197& 0.51595& 0.51994& 0.52392& 0.5279& 0.53188& 0.53586\\0.1& 0.53983& 0.5438& 0.54776& 0.55172& 0.55567& 0.55962& 0.56356& 0.56749& 0.57142& 0.57535\\0.2& 0.57926& 0.58317& 0.58706& 0.59095& 0.59483& 0.59871& 0.60257& 0.60642& 0.61026& 0.61409\\0.3& 0.61791& 0.62172& 0.62552& 0.6293& 0.63307& 0.63683& 0.64058& 0.64431& 0.64803& 0.65173\\0.4& 0.65542& 0.6591& 0.66276& 0.6664& 0.67003& 0.67364& 0.67724& 0.68082& 0.68439& 0.68793\\\hline0.5& 0.69146& 0.69497& 0.69847& 0.70194& 0.7054& 0.70884& 0.71226& 0.71566& 0.71904& 0.7224\\0.6& 0.72575& 0.72907& 0.73237& 0.73565& 0.73891& 0.74215& 0.74537& 0.74857& 0.75175& 0.7549\\0.7& 0.75804& 0.76115& 0.76424& 0.7673& 0.77035& 0.77337& 0.77637& 0.77935& 0.7823& 0.78524\\0.8& 0.78814& 0.79103& 0.79389& 0.79673& 0.79955& 0.80234& 0.80511& 0.80785& 0.81057& 0.81327\\0.9& 0.81594& 0.81859& 0.82121& 0.82381& 0.82639& 0.82894& 0.83147& 0.83398& 0.83646& 0.83891\\\hline1& 0.84134& 0.84375& 0.84614& 0.84849& 0.85083& 0.85314& 0.85543& 0.85769& 0.85993& 0.86214\\1.1& 0.86433& 0.8665& 0.86864& 0.87076& 0.87286& 0.87493& 0.87698& 0.879& 0.881& 0.88298\\1.2& 0.88493& 0.88686& 0.88877& 0.89065& 0.89251& 0.89435& 0.89617& 0.89796& 0.89973& 0.90147\\1.3& 0.9032& 0.9049& 0.90658& 0.90824& 0.90988& 0.91149& 0.91309& 0.91466& 0.91621& 0.91774\\1.4& 0.91924& 0.92073& 0.9222& 0.92364& 0.92507& 0.92647& 0.92785& 0.92922& 0.93056& 0.93189\\\hline1.5& 0.93319& 0.93448& 0.93574& 0.93699& 0.93822& 0.93943& 0.94062& 0.94179& 0.94295& 0.94408\\\end{array} \]Sei \(X\) eine \( N(\mu, \sigma=1)\)-verteilte Zufallsvariable für die \( P(X > 1) = 0.624\) gilt.

Wie groß ist \( \mu \)?

Verteilungsfunktion \( \Phi(x) \) der \(N(0,1)\) Verteilung \[ \begin{array}{l|llllllllll}x & ,00 & ,01 & ,02 & ,03 & ,04& ,05& ,06& ,07& ,08& ,09 \\ \hline0& 0.5& 0.50399& 0.50798& 0.51197& 0.51595& 0.51994& 0.52392& 0.5279& 0.53188& 0.53586\\0.1& 0.53983& 0.5438& 0.54776& 0.55172& 0.55567& 0.55962& 0.56356& 0.56749& 0.57142& 0.57535\\0.2& 0.57926& 0.58317& 0.58706& 0.59095& 0.59483& 0.59871& 0.60257& 0.60642& 0.61026& 0.61409\\0.3& 0.61791& 0.62172& 0.62552& 0.6293& 0.63307& 0.63683& 0.64058& 0.64431& 0.64803& 0.65173\\0.4& 0.65542& 0.6591& 0.66276& 0.6664& 0.67003& 0.67364& 0.67724& 0.68082& 0.68439& 0.68793\\\hline0.5& 0.69146& 0.69497& 0.69847& 0.70194& 0.7054& 0.70884& 0.71226& 0.71566& 0.71904& 0.7224\\0.6& 0.72575& 0.72907& 0.73237& 0.73565& 0.73891& 0.74215& 0.74537& 0.74857& 0.75175& 0.7549\\0.7& 0.75804& 0.76115& 0.76424& 0.7673& 0.77035& 0.77337& 0.77637& 0.77935& 0.7823& 0.78524\\0.8& 0.78814& 0.79103& 0.79389& 0.79673& 0.79955& 0.80234& 0.80511& 0.80785& 0.81057& 0.81327\\0.9& 0.81594& 0.81859& 0.82121& 0.82381& 0.82639& 0.82894& 0.83147& 0.83398& 0.83646& 0.83891\\\hline1& 0.84134& 0.84375& 0.84614& 0.84849& 0.85083& 0.85314& 0.85543& 0.85769& 0.85993& 0.86214\\1.1& 0.86433& 0.8665& 0.86864& 0.87076& 0.87286& 0.87493& 0.87698& 0.879& 0.881& 0.88298\\1.2& 0.88493& 0.88686& 0.88877& 0.89065& 0.89251& 0.89435& 0.89617& 0.89796& 0.89973& 0.90147\\1.3& 0.9032& 0.9049& 0.90658& 0.90824& 0.90988& 0.91149& 0.91309& 0.91466& 0.91621& 0.91774\\1.4& 0.91924& 0.92073& 0.9222& 0.92364& 0.92507& 0.92647& 0.92785& 0.92922& 0.93056& 0.93189\\\hline1.5& 0.93319& 0.93448& 0.93574& 0.93699& 0.93822& 0.93943& 0.94062& 0.94179& 0.94295& 0.94408\\\end{array} \]Sei \(X\) eine \( N(\mu , \sigma=0.75)\)-verteilte Zufallsvariable für die \( P(X < 0.4) = 0.766\) gilt.

Wie groß ist \( \mu \)?

Verteilungsfunktion \( \Phi(x) \) der \(N(0,1)\) Verteilung \[ \begin{array}{l|llllllllll}x & ,00 & ,01 & ,02 & ,03 & ,04& ,05& ,06& ,07& ,08& ,09 \\ \hline0& 0.5& 0.50399& 0.50798& 0.51197& 0.51595& 0.51994& 0.52392& 0.5279& 0.53188& 0.53586\\0.1& 0.53983& 0.5438& 0.54776& 0.55172& 0.55567& 0.55962& 0.56356& 0.56749& 0.57142& 0.57535\\0.2& 0.57926& 0.58317& 0.58706& 0.59095& 0.59483& 0.59871& 0.60257& 0.60642& 0.61026& 0.61409\\0.3& 0.61791& 0.62172& 0.62552& 0.6293& 0.63307& 0.63683& 0.64058& 0.64431& 0.64803& 0.65173\\0.4& 0.65542& 0.6591& 0.66276& 0.6664& 0.67003& 0.67364& 0.67724& 0.68082& 0.68439& 0.68793\\\hline0.5& 0.69146& 0.69497& 0.69847& 0.70194& 0.7054& 0.70884& 0.71226& 0.71566& 0.71904& 0.7224\\0.6& 0.72575& 0.72907& 0.73237& 0.73565& 0.73891& 0.74215& 0.74537& 0.74857& 0.75175& 0.7549\\0.7& 0.75804& 0.76115& 0.76424& 0.7673& 0.77035& 0.77337& 0.77637& 0.77935& 0.7823& 0.78524\\0.8& 0.78814& 0.79103& 0.79389& 0.79673& 0.79955& 0.80234& 0.80511& 0.80785& 0.81057& 0.81327\\0.9& 0.81594& 0.81859& 0.82121& 0.82381& 0.82639& 0.82894& 0.83147& 0.83398& 0.83646& 0.83891\\\hline1& 0.84134& 0.84375& 0.84614& 0.84849& 0.85083& 0.85314& 0.85543& 0.85769& 0.85993& 0.86214\\1.1& 0.86433& 0.8665& 0.86864& 0.87076& 0.87286& 0.87493& 0.87698& 0.879& 0.881& 0.88298\\1.2& 0.88493& 0.88686& 0.88877& 0.89065& 0.89251& 0.89435& 0.89617& 0.89796& 0.89973& 0.90147\\1.3& 0.9032& 0.9049& 0.90658& 0.90824& 0.90988& 0.91149& 0.91309& 0.91466& 0.91621& 0.91774\\1.4& 0.91924& 0.92073& 0.9222& 0.92364& 0.92507& 0.92647& 0.92785& 0.92922& 0.93056& 0.93189\\\hline1.5& 0.93319& 0.93448& 0.93574& 0.93699& 0.93822& 0.93943& 0.94062& 0.94179& 0.94295& 0.94408\\\end{array} \]Sei \(X\) eine \( N(0, \sigma = 1)\)-verteilte Zufallsvariable

Wie groß ist die Wahrscheinlichkeit \[ P(-0.763< X < 1.419) ?\]

Verteilungsfunktion \( \Phi(x) \) der \(N(0,1)\) Verteilung \[ \begin{array}{l|llllllllll}x & ,00 & ,01 & ,02 & ,03 & ,04& ,05& ,06& ,07& ,08& ,09 \\ \hline0& 0.5& 0.50399& 0.50798& 0.51197& 0.51595& 0.51994& 0.52392& 0.5279& 0.53188& 0.53586\\0.1& 0.53983& 0.5438& 0.54776& 0.55172& 0.55567& 0.55962& 0.56356& 0.56749& 0.57142& 0.57535\\0.2& 0.57926& 0.58317& 0.58706& 0.59095& 0.59483& 0.59871& 0.60257& 0.60642& 0.61026& 0.61409\\0.3& 0.61791& 0.62172& 0.62552& 0.6293& 0.63307& 0.63683& 0.64058& 0.64431& 0.64803& 0.65173\\0.4& 0.65542& 0.6591& 0.66276& 0.6664& 0.67003& 0.67364& 0.67724& 0.68082& 0.68439& 0.68793\\\hline0.5& 0.69146& 0.69497& 0.69847& 0.70194& 0.7054& 0.70884& 0.71226& 0.71566& 0.71904& 0.7224\\0.6& 0.72575& 0.72907& 0.73237& 0.73565& 0.73891& 0.74215& 0.74537& 0.74857& 0.75175& 0.7549\\0.7& 0.75804& 0.76115& 0.76424& 0.7673& 0.77035& 0.77337& 0.77637& 0.77935& 0.7823& 0.78524\\0.8& 0.78814& 0.79103& 0.79389& 0.79673& 0.79955& 0.80234& 0.80511& 0.80785& 0.81057& 0.81327\\0.9& 0.81594& 0.81859& 0.82121& 0.82381& 0.82639& 0.82894& 0.83147& 0.83398& 0.83646& 0.83891\\\hline1& 0.84134& 0.84375& 0.84614& 0.84849& 0.85083& 0.85314& 0.85543& 0.85769& 0.85993& 0.86214\\1.1& 0.86433& 0.8665& 0.86864& 0.87076& 0.87286& 0.87493& 0.87698& 0.879& 0.881& 0.88298\\1.2& 0.88493& 0.88686& 0.88877& 0.89065& 0.89251& 0.89435& 0.89617& 0.89796& 0.89973& 0.90147\\1.3& 0.9032& 0.9049& 0.90658& 0.90824& 0.90988& 0.91149& 0.91309& 0.91466& 0.91621& 0.91774\\1.4& 0.91924& 0.92073& 0.9222& 0.92364& 0.92507& 0.92647& 0.92785& 0.92922& 0.93056& 0.93189\\\hline1.5& 0.93319& 0.93448& 0.93574& 0.93699& 0.93822& 0.93943& 0.94062& 0.94179& 0.94295& 0.94408\\\end{array} \]Sei \(X\) eine \( N(\mu = 0.61, \sigma )\)-verteilte Zufallsvariable für die \( P(X < 1.21) = 0.516\) gilt.

Wie groß ist \( \sigma \)?