Question

Task 1
Calculate the following probabilities:
a) P (| X | <1.5), P (X> 2), P (X> -1), X: N (0, 1) -distributed
b) P (-2≤X <7), P (-5≤X≤2), P (X> 0), X: N (1, 9) -distributed
c) σ * 2, if X: N (2, σ * 2) distributed with P (0 <X <4) = 0.68269.
d) μ, if X: N (μ, 16) distributed with P (X <7) = 0.3265.

exercise 2
Experience shows that the size of the resistors originating from a certain production is a normally distributed random variable with the parameters μ = 990Ω and σ = 20Ω. All resistances,
which are not within the tolerance limits of 950Ω to 1050Ω are to be regarded as rejects.
a) What percentage of resistances produced are rejects?
b) By changing the technology it was possible to increase the mean value to 1000Ω. Which benefit arises?

c) For installation in precision equipment, resistors of 990Ω to 1010Ω are required. What is the maximum allowed value of σ (μ = 1000) if 90% of the resistors are to meet the conditions for installation?

Task 3
The filling quantity of automatically filled spray bottles is a normally distributed random variable with the known parameters μ = 150cm * 3 and σ = 3.5cm * 3. In quality control, a bottle is discarded if it contains less than 146cm3. The daily production of spray bottles amounts to 4000 pieces.

a) How many spray bottles can be put on average on average daily?
b) At which new value μ would the filling machine have to be adjusted, so that on average only 1% of the spray bottles are complained about during the quality control?

Task 4
A wear part with exponentialverteilter life is built into an aggregate and have
the failure rate λ = 5.2 · 10 * -3h * -1
,
a) How many such parts do you need on average in one year (365 days)?
b) How many such parts do you have to reserve so that they are 99% safe enough to run the unit in one year?
c) Question as in b) but with 50% certainty.

Answer 1