Problem 27: Find the value of determinant of the diagonal matrix

\begin{align*} \Delta =\left| \begin{array}{cccc} a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& d \end{array}\right| \end{align*}

Solution:

On expanding $\Delta$ along the $R_1$, we get

\begin{align*} \Delta =a\left| \begin{array}{ccc} b& 0& 0\\ 0& c& 0\\ 0& 0& d\\ \end{array}\right| \end{align*} \begin{align*} \Delta =ab\left| \begin{array}{ccc} c& 0\\ 0& d\\ \end{array}\right| \end{align*} \begin{align*} \Delta =abcd. \end{align*}

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Problem 105: If a bank receives on the average α = 6 bad checks per day, what are the probabilities that it will receive (a) 4 bad checks on any given day?
(b) 10 bad checks over any 2 consecutive days?

Problem 105: If a bank receives on the average α = 6 bad checks per day, what are the probabilities that it will receive (a) 4 bad checks on any given day? (b) 10 bad checks over any 2 consecutive days? Solution: Poisson distribution, $f(x;\lambda )=\frac{{\lambda }^{x}{e}^{-\lambda }}{x!}$ a. 4 bad checks on any given day Given that $x=4, \lambda = \alphaT = 6\dot 1 = 6$ $f(4;6 )=\frac{{6 }^{4}{e}^{-6 }}{4!} = 0.134$ b. 10 bad checks over any 2 consecutive days Given that $x=10, \lambda = \alphaT = 6\dot 2 = 12$ $f(10;12 )=\frac{{12 }^{10}{e}^{-12 }}{10!} = 0.104837$

Problem 98: If 6 of 18 new buildings in a city violate the building code, what is the probability that a building inspector, who randomly selects 4 of the new buildings for inspection, will catch
a. none of the buildings that violate the building code?
b. 1 of the new buildings that violate the building code?
c. 2 of the new buildings that violate the building code?
d. at least 3 of the new buildings that violate the building code?

Problem 98: If 6 of 18 new buildings in a city violate the building code, what is the probability that a building inspector, who randomly selects 4 of the new buildings for inspection, will catch a. none of the buildings that violate the building code? b. 1 of the new buildings that violate the building code? c. 2 of the new buildings that violate the building code? d. at least 3 of the new buildings that violate the building code? Solution: a. none of the buildings that violate the building code? Given that $x=0,n=4,a=6, N=18$ Hypergeometric distribution, $h(x;n,a,N)=\frac{\left(\begin{array}{c}a\\ x\end{array}\right)\left(\begin{array}{c}N-a\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$ $P(x=0) = h(0;4,6,18)=\frac{\left(\begin{array}{c}6\\ 0\end{array}\right)\left(\begin{array}{c}18-6\\ 4-0\end{array}\right)}{\left(\begin{array}{c}18\\ 4\end{array}\right)} = 0.16176.$ b. 1 of the new buildings that violate the building code? Given ...

Problem 101: Among the 300 employees of a company, 240 are union members, while the others are not, If 8 of the employees are chosen to serve on the administrative committee, find the probability that 5 of them will be union member while the others are not.
a. the formula for the hypergeometric distribution;
b. the formula for the binomial distribution as an approximation.

Problem 101: Among the 300 employees of a company, 240 are union members, while the others are not, If 8 of the employees are chosen to serve on the administrative committee, find the probability that 5 of them will be union member while the others are not. a. the formula for the hypergeometric distribution; b. the formula for the binomial distribution as an approximation. Solution: a. the formula for the hypergeometric distribution Given that $x=5,n=8,a=240, N=300$ Hypergeometric distribution, $h(x;n,a,N)=\frac{\left(\begin{array}{c}a\\ x\end{array}\right)\left(\begin{array}{c}N-a\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$ $P(x=5) = h(5;8,240,300)=\frac{\left(\begin{array}{c}240\\ 5\end{array}\right)\left(\begin{array}{c}300-240\\ 8-5\end{array}\right)}{\left(\begin{array}{c}300\\ 8\end{array}\right)} = 0.1470.$ b. the formula for the binomial distribution as an approximation? Given that $x=5,n=8,p = \frac{240}{300}$ $P(x= 5) =...

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