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Exercises

Exercises. Normal distribution. Exercise s. The surface area under the density function of th e standard normal distribution to the left of a given value u is denoted as Φ (u ) . It is the probability to observe a value less than or equal to u , i .e. P(U ≤ u)= Φ (u ) .

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Exercises

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  1. Exercises Normal distribution

  2. Exercises • The surfaceareaunderthedensityfunctionofthestandard normal distributiontotheleftof a givenvalueuisdenotedasΦ(u).Itistheprobabilitytoobserve a valuelessthanorequaltou, i.e. P(U ≤ u)= Φ(u). • Someidentities: • P(U > u)=1- P(U ≤ u) • Φ(-u)=1-Φ(u)

  3. Exercises • ComputeΦ(u) forthefollowingvaluesofu: • u=1.55 • u=-0.86 • ComputeP(U > u) forthefollowingvaluesofu: • u=1.11 • u=-0.45 • ComputeP(for the following values ofu: • =1.28, =1.96 • =-1.00, =2.00

  4. Exercises • Computethe25%-, 50%- und 75%-quantileofthestandard normal distribution. • Supposethatthesystolicbloodpressureobtainedfroma sample ofpatientsisnormallydistributedwithexpecationμ=120 andvarianceσ²=100. Computethefollowing. • P(X110) • P(X>140) • P(X(130,140]) d) The 90%- quantile

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