Question

Let X be a random variable having an unknown distribution with mean m=8 and variance s²=9 (standard deviation s =3). Find the following probability: (a) P(-4 <X< 20) (b) P(|X-8| ³ 6)

Answer 1

By Chebyshev’s Theorem for any numerical data set,

at least $3/4$  of the data lie within two standard deviations of the mean, that is, in the interval with endpoints $\bar{x}\pm2s$ for samples,

at least $8/9$  of the data lie within three standard deviations of the mean, that is, in the interval with endpoints $\bar{x}\pm3s$ for samplesm

at least $1-1/k^2$ of the data lie within $k$ standard deviations of the mean, that is, in the interval with endpoints $x\pm ks$ for samples, where $k$ is any positive whole number that is greater than $1.$

Given $\bar{x}=m=8, s=\sqrt{s^2}=3.$

(a)

$$P(-4<X<20)=P(8-4(3)<X<8+4(3))$$

$$=1-\dfrac{1}{4^2}=\dfrac{15}{16}$$

(b)

$$P(|X-8|<6)=P(8-2(3)<X<8+2(3))$$

$$=\dfrac{8}{9}$$