The property of a probability density function: integral over all space is equal to 1.
f(x)=10e−10(x−60),x>6
is not a probability density function, because the property fails.
Let's try to fix it. Let the probability density function be
f(x)=10e−10(x−6),x>6.
Then the property succeeds:
−∞∫∞f(x)dx=6∫∞10e−10(x−6)dx=−e−10(x−6)∣6∞=0−(−1)=1
Solution
M(X)=−∞∫∞x⋅f(x)dx=6∫∞10x⋅e−10(x−6)dx==x⋅(−e−10(x−6))∣6∞−6∫∞−e−10(x−6)dx==0−(−6)−0.1(e−10(x−6))∣6∞=6−(0−0.1)=6.1
M(X2)=−∞∫∞x2⋅f(x)dx=6∫∞10x2⋅e−10(x−6)dx==x2⋅(−e−10(x−6))∣6∞−6∫∞2x(−e−10(x−6))dx==0−(−36)−((−0.2)⋅6∫∞10x⋅e−10(x−6)dx)=36+0.2⋅6.1=37.22
D(X)=M(X2)−M2(X)=37.22−37.21=0.01
Answer
Mean = 6.1
Variance = 0.01
6.1∫∞f(x)dx=6.1∫∞10e−10(x−6)dx=−e−10(x−6)∣6.1∞=0−(−e−1)=1/e≈0.368
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