The probability density function of a continuous random variable X is given below:
π(π₯) = {
1.25(1 β π₯
4
) , 0 < π₯ < 1
0 ππ‘βπππ€ππ π
a. Find P(0.4< X<0.9)
b. Compute the expected value and variance of X.
a:P(0.4<X<0.9)=β«0.40.91.25(1βx4)dx=1.25(0.5β0.95β0.455)=0.479938b:EX=β«01xβ 1.25(1βx4)dx=1.25(12β16)=0.416667EX2=β«01x2β 1.25(1βx4)dx=1.25(13β17)=0.238095Var(X)=0.238095β0.4166672=0.0644836a:\\P\left( 0.4<X<0.9 \right) =\int_{0.4}^{0.9}{1.25\left( 1-x^4 \right) dx}=1.25\left( 0.5-\frac{0.9^5-0.4^5}{5} \right) =0.479938\\b:\\EX=\int_0^1{x\cdot 1.25\left( 1-x^4 \right) dx}=1.25\left( \frac{1}{2}-\frac{1}{6} \right) =0.416667\\EX^2=\int_0^1{x^2\cdot 1.25\left( 1-x^4 \right) dx}=1.25\left( \frac{1}{3}-\frac{1}{7} \right) =0.238095\\Var\left( X \right) =0.238095-0.416667^2=0.0644836a:P(0.4<X<0.9)=β«0.40.9β1.25(1βx4)dx=1.25(0.5β50.95β0.45β)=0.479938b:EX=β«01βxβ 1.25(1βx4)dx=1.25(21ββ61β)=0.416667EX2=β«01βx2β 1.25(1βx4)dx=1.25(31ββ71β)=0.238095Var(X)=0.238095β0.4166672=0.0644836
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