Question

A commuter must pass through 5 traffic lights on her way to work and will have to stop at each one that is red. She estimates the probability model for the number of red lights she hits as shown below.X = # of red 0 1 2 3 4 5P(X = x) 0.06 0.25 0.35 0.15 0.13 0.06(a) Compute the mean, or expected value, of the random variable X. (Round to one decimal place as needed.)(b) Compute the standard deviation of the random variable X. (Round the final answer to one decimal place as needed. Round all intermediate values to three decimal places as needed.)

Answer 1

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)         $X^{2}$         $X^{2} * P(X=x)$

              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 = $\sum X*P(X=x)$

                                                                                           = 2.22

(b) Variance of random variable X = $\sum X^{2} *P(X=x) - (\sum X*P(X=x))^{2}$

                                                        = 6.58 - $2.22^{2}$ = 1.6516

So, standard deviation = $\sqrt{Variance}$ = $\sqrt{1.6516}$ = 1.285

Answer 2

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:

$\mu=E(X)=\sum x.P(X=x)$

The formula to compute the standard deviation of X is:

$\sigma=\sqrt{E(X^{2})-(E(X))^{2}}$

The formula of E (X²) is:

$E(X^{2})=\sum x^{2}.P(X=x)$

(a)

Compute the expected value of X as follows:

$E(X)=\sum x.P(X=x)\\=(0\times0.06)+(1\times0.25)+(2\times0.35)+(3\times0.15)+(4\times0.13)+(5\times0.06)\\=2.22\\\approx2.2$

Thus, the mean or expected value of X is 2.2.

(b)

Compute the value of E (X²) as follows:

$E(X^{2})=\sum x^{2}.P(X=x)\\=(0^{2}\times0.06)+(1^{2}\times0.25)+(2^{2}\times0.35)+(3^{2}\times0.15)+(4^{2}\times0.13)+(5^{2}\times0.06)\\=6.58$

Compute the standard deviation of X as follows:

$\sigma=\sqrt{E(X^{2})-(E(X))^{2}}\\=\sqrt{6.58-(2.22)^{2}}\\=\sqrt{1.6516}\\=1.285\\\approx1.3$

Thus, the standard deviation of X is 1.3.