Respuesta :
Answer:
(a) The mean or expected value of X is 2.2.
(b) The standard deviation of X is 1.3.
Step-by-step explanation:
Let X = number of times the traffic light is red when a commuter passes through the traffic lights.
The probability distribution of X id provided.
The formula to compute the mean or expected value of X is:
[tex]\mu=E(X)=\sum x.P(X=x)[/tex]
The formula to compute the standard deviation of X is:
[tex]\sigma=\sqrt{E(X^{2})-(E(X))^{2}}[/tex]
The formula of E (X²) is:
[tex]E(X^{2})=\sum x^{2}.P(X=x)[/tex]
(a)
Compute the expected value of X as follows:
[tex]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[/tex]
Thus, the mean or expected value of X is 2.2.
(b)
Compute the value of E (X²) as follows:
[tex]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[/tex]
Compute the standard deviation of X as follows:
[tex]\sigma=\sqrt{E(X^{2})-(E(X))^{2}}\\=\sqrt{6.58-(2.22)^{2}}\\=\sqrt{1.6516}\\=1.285\\\approx1.3[/tex]
Thus, the standard deviation of X is 1.3.
Answer:
(a) Expected value, of the random variable X = 2.22
(b) Standard deviation of the random variable X = 1.285
Step-by-step explanation:
We are given the probability model for the number of red lights she hits as shown below;
No. of red lights (X) P(X = x) X*P(X = x) [tex]X^{2}[/tex] [tex]X^{2} * P(X=x)[/tex]
0 0.06 0 0 0
1 0.25 0.25 1 0.25
2 0.35 0.70 4 1.40
3 0.15 0.45 9 1.35
4 0.13 0.52 16 2.08
5 0.06 0.30 25 1.50
∑P(X=x) = 1 Total = 2.22 Total = 6.58
(a) Mean, or expected value, of the random variable X = [tex]\sum X*P(X=x)[/tex]
= 2.22
(b) Variance of random variable X = [tex]\sum X^{2} *P(X=x) - (\sum X*P(X=x))^{2}[/tex]
= 6.58 - [tex]2.22^{2}[/tex] = 1.6516
So, standard deviation = [tex]\sqrt{Variance}[/tex] = [tex]\sqrt{1.6516}[/tex] = 1.285
