Problem 68: V is the vector space of all real-valued functions defined on the interval $(-\infty,\infty)$, and S is the subset of V consisting of all functions satisfying $f(-x) = f(x)$ for all $(-\infty,\infty)$

Problem 68: V is the vector space of all real-valued functions defined on the interval $(-\infty,\infty)$, and S is the subset of V consisting of all functions satisfying $f(-x) = f(x)$ for all $(-\infty,\infty)$. Determine whether it is a subspace of the given vector space $\V$.

Solution:
$S= \{f \in V : f(-x) = f(x)$ for all $x \in R\}$,
where V is the vector space of all real-valued functions defined on $(-\infty,\infty)$.
If $f, g \in S$, then
$(f + g)(-x) = f(-x) + g(-x) = f(x)+g(x) = (f+g)(x)$,
which shows that $f + g \in S$.
and If $c \in R$, then
$(cf)(-x) = cf(-x) = cf(x) = (cf)x$,
which shows that $cf \in S$.
Therefore S is a subspace of V

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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 ...

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a. the formula for the hypergeometric distribution;
b. the formula for the binomial distribution as an approximation.

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