Answer:
a.[tex]P(x\geq 75)\leq 0.67[/tex]
b.P(40<x<60)[tex]\geq[/tex]0.75
Step-by-step explanation:
We are given that
Mean =E(X)=50
a.We have to find the probability when x greater than or equal to 75.
Markovs inequality
[tex]P(x\geq k)\leq \frac{E(X)}{k}[/tex]
By using Markovs inequality and substitute k=75
[tex]P(x\geq 75)\leq \frac{E(x)}{75}[/tex]
[tex]P(x\geq 75)\leq \frac{50}{75}=\frac{2}{3}=0.67[/tex]
[tex]P(x\geq 75)\leq 0.67[/tex]
b.We have to find P(40<x<60)
Variance=[tex]\sigma=25[/tex]
Chebyshev's inequality:[tex]P(\mid X-E(X)\geq k)\leq \frac{\sigma^2}{k^2}[/tex]
Because 50+10=60 and 50-10=40
Therefore, k=10
By using Chebyshev's inequality and substitute k=10
because 50+10=60 and 50-10=40
[tex]P(\mid x-50\mid \geq 10)\leq \frac{(25)^2}{10^2}[/tex]
[tex]P(\mid x-50\mid \geq 10)\leq \frac{1}{4}[/tex]
[tex]P(\mid x-50\mid <10)\geq 1-\frac{1}{4}=\frac{3}{4}=0.75[/tex]
Hence, P(40<x<60)[tex]\geq[/tex]0.75
