Question

5. The reading given by a thermometer in ice water (actual temperature 0° C) is a random variable with pdf otherwise where k is a constant. (a) Find k b) Find the probability that the reading is within 25 (e) Find the expected reading. (d) Find the median reading. (e) Find the standard deviation of the reading. C of the actual temperature

Answer 1

a) for this to be valid: $\int_{-1}^{2}$ f(x)dx must be 1.

$\int_{-1}^{2}$ f(x)dx =$\int_{-1}^{2}$ k*(4-x²) dx =k(4x-x³/3) |²_-1 =k*9=1

k=1/9

b)

P(-0.25<X<0.25)=$\int_{-0.25}^{0.25}$ f(x)dx =$\int_{-0.25}^{0.25}$ (1/9)*(4-x²) dx =(1/9)(4x-x³/3) |^0.25_-0.25 =0.221065

c)

expected value =$\int_{-1}^{2}$ xf(x)dx =$\int_{-1}^{2}$ (1/9)*(4x-x³) dx =(1/9)(4x²/2-x⁴/4) |²_-1 =0.25

d)

let median be at x=m

hence $\int_{-1}^{m}$ f(x)dx =$\int_{-1}^{m}$ (1/9*(4-x²) dx =(1/9)(4x-x³/3) |^m_-1 =(1/9)*(4m-m³/3)-(-0.40741)=0.5

solving above median=0.2091

e)

E(X²)=$\int_{-1}^{2}$ x²f(x)dx =$\int_{-1}^{2}$ (1/9)*(4x²-x⁴) dx =(1/9)(4x³/3-x⁵/5) |²_-1 =0.6

hence std deviation =sqrt(E(X²)-(E(X))²) =sqrt(0.6-0.25²) =0.7331