Answer:
(a) The mean or expected value of <em>X </em>is 2.2.
(b) The standard deviation of <em>X</em> is 1.3.
Step-by-step explanation:
Let <em>X</em> = number of times the traffic light is red when a commuter passes through the traffic lights.
The probability distribution of <em>X</em> id provided.
The formula to compute the mean or expected value of <em>X </em>is:
![\mu=E(X)=\sum x.P(X=x)](https://tex.z-dn.net/?f=%5Cmu%3DE%28X%29%3D%5Csum%20x.P%28X%3Dx%29)
The formula to compute the standard deviation of <em>X </em>is:
![\sigma=\sqrt{E(X^{2})-(E(X))^{2}}](https://tex.z-dn.net/?f=%5Csigma%3D%5Csqrt%7BE%28X%5E%7B2%7D%29-%28E%28X%29%29%5E%7B2%7D%7D)
The formula of E (X²) is:
![E(X^{2})=\sum x^{2}.P(X=x)](https://tex.z-dn.net/?f=E%28X%5E%7B2%7D%29%3D%5Csum%20x%5E%7B2%7D.P%28X%3Dx%29)
(a)
Compute the expected value of <em>X</em> as follows:
![E(X)=\sum x.P(X=x)\\=(0\times0.06)+(1\times0.25)+(2\times0.35)+(3\times0.15)+(4\times0.13)+(5\times0.06)\\=2.22\\\approx2.2](https://tex.z-dn.net/?f=E%28X%29%3D%5Csum%20x.P%28X%3Dx%29%5C%5C%3D%280%5Ctimes0.06%29%2B%281%5Ctimes0.25%29%2B%282%5Ctimes0.35%29%2B%283%5Ctimes0.15%29%2B%284%5Ctimes0.13%29%2B%285%5Ctimes0.06%29%5C%5C%3D2.22%5C%5C%5Capprox2.2)
Thus, the mean or expected value of <em>X </em>is 2.2.
(b)
Compute the value of E (X²) as follows:
![E(X^{2})=\sum x^{2}.P(X=x)\\=(0^{2}\times0.06)+(1^{2}\times0.25)+(2^{2}\times0.35)+(3^{2}\times0.15)+(4^{2}\times0.13)+(5^{2}\times0.06)\\=6.58](https://tex.z-dn.net/?f=E%28X%5E%7B2%7D%29%3D%5Csum%20x%5E%7B2%7D.P%28X%3Dx%29%5C%5C%3D%280%5E%7B2%7D%5Ctimes0.06%29%2B%281%5E%7B2%7D%5Ctimes0.25%29%2B%282%5E%7B2%7D%5Ctimes0.35%29%2B%283%5E%7B2%7D%5Ctimes0.15%29%2B%284%5E%7B2%7D%5Ctimes0.13%29%2B%285%5E%7B2%7D%5Ctimes0.06%29%5C%5C%3D6.58)
Compute the standard deviation of <em>X</em> as follows:
![\sigma=\sqrt{E(X^{2})-(E(X))^{2}}\\=\sqrt{6.58-(2.22)^{2}}\\=\sqrt{1.6516}\\=1.285\\\approx1.3](https://tex.z-dn.net/?f=%5Csigma%3D%5Csqrt%7BE%28X%5E%7B2%7D%29-%28E%28X%29%29%5E%7B2%7D%7D%5C%5C%3D%5Csqrt%7B6.58-%282.22%29%5E%7B2%7D%7D%5C%5C%3D%5Csqrt%7B1.6516%7D%5C%5C%3D1.285%5C%5C%5Capprox1.3)
Thus, the standard deviation of <em>X</em> is 1.3.