Question

1. the intersection of river and 1st avenue averages 7.0 accidents per month. let x = the number of accidents in a randomly selected month. find:

a. p(x = 5)
b. p(x < 2)
c. p(x > 3)
d. p(x = 8)
e. p(x < 10)
f. p(x ≥ 5)

Explanation:

Step1: Identify the distribution

Since we are dealing with the number of events (accidents) in a fixed interval (a month) and the average number of events is given, \(X\) follows a Poisson distribution. The probability mass function of a Poisson distribution is \(P(X = k)=\frac{\lambda^{k}e^{-\lambda}}{k!}\), where \(\lambda\) is the average number of events. Here, \(\lambda = 7\).

Step2: Calculate \(P(X = 5)\)

Substitute \(k = 5\) and \(\lambda=7\) into the formula:

$$
LATEXBLOCK0
$$

Step3: Calculate \(P(X\lt2)=P(X = 0)+P(X = 1)\)

For \(k = 0\):

$$
LATEXBLOCK1
$$

For \(k = 1\):

$$
LATEXBLOCK2
$$

\(P(X\lt2)=0.00091 + 0.00637=0.00728\)

Step4: Calculate \(P(X\gt3)=1 - P(X\leq3)=1-(P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3))\)

For \(k = 2\):

$$
LATEXBLOCK3
$$

For \(k = 3\):

$$
LATEXBLOCK4
$$

\(P(X\leq3)=0.00091+0.00637 + 0.022295+0.05202=0.081595\)
\(P(X\gt3)=1 - 0.081595=0.918405\)

Step5: Calculate \(P(X = 8)\)

Substitute \(k = 8\) and \(\lambda = 7\) into the formula:

$$
LATEXBLOCK5
$$

Step6: Calculate \(P(X\lt10)=1 - P(X\geq10)\)

\(P(X\geq10)=1-(P(X = 0)+P(X = 1)+\cdots+P(X = 9))\)
Using a Poisson calculator or cumulative Poisson formula:
\(P(X\lt10)\approx0.8305\)

Step7: Calculate \(P(X\geq5)=1 - P(X\lt5)=1-(P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4))\)

For \(k = 4\):

$$
LATEXBLOCK6
$$

\(P(X\lt5)=0.00091+0.00637+0.022295+0.05202+0.09104 = 0.172635\)
\(P(X\geq5)=1 - 0.172635=0.827365\)

Answer:

A. \(P(X = 5)\approx0.1275\)
B. \(P(X\lt2)\approx0.0073\)
C. \(P(X\gt3)\approx0.9184\)
D. \(P(X = 8)\approx0.1301\)
E. \(P(X\lt10)\approx0.8305\)
F. \(P(X\geq5)\approx0.8274\)