Assuming that the total cholesterol in a population (in mmol /L) X ∼ N(4.2, 0.8), and that total cholesterol is considered high if it exceeds 5.6 mmol/L, find the probability that the total cholesterol is high.

CDF[NormalDistribution[0,1],x]

P(X>5.6)=1-(1/2)Erfc[-(((5.6-4.2)/0.8)/(Sqrt[2]))]

1-(1/2)Erfc[-(((5.6-4.2)/0.8)/(Sqrt[2]))]

Assuming that the height of a bus door is designed in such a way that the chance of a man touching the top of the door is less than 0.01, and given that the height of a man (in centimeters) X ∼ N(175, 6), how is the height of the bus door designed?

let the height of boor be h

P (X > h) <= 0.01 is the same as P (X <= h) <= 0.99

P (X <= h)=ϕ(((h-μ)/σ))=(1/2)Erfc[-(((h-175)/6)/(Sqrt[2]))]>=0.99

Reduce[(1/2)Erfc[-(((h-175)/6)/(Sqrt[2]))]>=99/100,h]

After looking up in the table, we can find that ϕ(2.33)=0.99
So h = 2.33*σ + μ = 2.33*6 + 175 = 189

Let X ∼ N(0, 1), Y ∼ N(0, 1), X, Y be independent of each other, and find Z = X^2 + Y^2.

(*Define the random variables*)
X=NormalDistribution[0,1];
Y=NormalDistribution[0,1];
(*Define the new variable Z=X^2+Y^2*)
Z=TransformedDistribution[x^2+y^2,{x\[Distributed]X,y\[Distributed]Y}];
(*Display the distribution of Z*)
Z

PDF[ChiSquareDistribution[2],x]