Question

Suppose that it is known that the number of items produced at a factory per week is a random variable X with mean 50. a) What can we say about the probability X greater than or equal to 75? b) Suppose that the variance of X is 25. What can we say about P(40 < x < 60)?

Answer 1

a.
`P(x\geq 75)\leq 0.67`

b.P(40<x<60)
`\geq`0.75

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

`P(x\geq k)\leq (E(X))/(k)`

By using Markovs inequality and substitute k=75

`P(x\geq 75)\leq (E(x))/(75)`

`P(x\geq 75)\leq (50)/(75)=(2)/(3)=0.67`

`P(x\geq 75)\leq 0.67`

b.We have to find P(40<x<60)

Variance=
`\sigma=25`

Chebyshev's inequality:
`P(\mid X-E(X)\geq k)\leq (\sigma^2)/(k^2)`

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

`P(\mid x-50\mid \geq 10)\leq ((25)^2)/(10^2)`

`P(\mid x-50\mid \geq 10)\leq (1)/(4)`

`P(\mid x-50\mid <10)\geq 1-(1)/(4)=(3)/(4)=0.75`

Hence, P(40<x<60)
`\geq`0.75