Problem 78: Checking for Nonnegativity and total probability equals one

Problem 78: Check Whether the following can serve as probability distributions:

$(a)$ $f(x) = \frac{x-2}{2}$ for $x=1,2,3,4$
$(b)$ $h(x) = \frac{x^2}{25}$ for $x=0,1,2,3,4$
$(c)$ $f(x) = \frac{x+2}{25}$ for $x=1,2,3,4,5$

Solution:

The probability distribution always satisfies the conditions

$f(x)\ge 0$ and $\sum _{all\;x}^{}f(x)\,=1$

(a) This fucntion cannot serve as a probability distribution because $f(1)$ is negative.

(b) This fucntion cannot serve as a probability distribution because the sum of the five probabilities is 6/5 and not 1.

(b) This fucntion can serve as a probability distribution because the sum of the five probabilities is $\frac{3}{25}+\frac{4}{25}+\frac{4}{25}+\frac{6}{25}+\frac{7}{25} = \frac{25}{25} = 1$.

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Problem 85: A manufacturer of external hard drives claims that only 10% of his drives require repairs within the warranty peroid of 12 months.

Problem 85: A manufacturer of external hard drives claims that only 10% of his drives require repairs within the warranty peroid of 12 months. a. If 5 of 20 of his drives required repairs within the first year, does this tend to support or refuse the claim? b. If 3 of 20 of his drives required repairs within the first year, does this tend to support or refuse the claim? Solution: a. If 5 of 20 of his drives required repairs within the first year Given that $n=20,$ and $p=0.3$ $\sum _{x=5}^{20}b(x;n,p)=\sum _{x=5}^{20}b(x;20,0.1)=1-\sum _{x=0}^{4}b(x;20,0.1)$ $ = 1 - [b(0;20,0.1)+b(1;20,0.1)+b(2;20,0.1)+b(3;20,0.1)+b(4;20,0.1)]$ $b(x;n,p)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}(1-p)^{n-x}$ $b(0;20,0.1) = \left(\begin{array}{c}20\\ 0\end{array}\right){0.1}^{0}(1-0.1)^{20-0} = 0.121577$ $b(1;20,0.1) = \left(\begin{array}{c}20\\ 1\end{array}\right){0.1}^{1}(1-0.1)^{20-1} = 0.27017$ $b(2;20,0.1) = \left(\begin{array}{c}20\\ 2\end{array}\right){0.1}^{...

Problem 56: Let $S = \{x \in \R^3 : x = (r-2s,3r+s,s), r,s \in \R \}$. Establish that $S$ is a subspace of $\R^3$.

Problem 56: Let $S = \{x \in \R^3 : x = (r-2s,3r+s,s), r,s \in \R \}$. Establish that $S$ is a subspace of $\R^3$. Solution: $S = \{x \in \R^3 : x = (r-2s,3r+s,s), r,s \in \R \}$. S is certainly nonempty. Let $x, y \in S$. Then for some $r, s, u,v \in \R$, $x = (r-2s,3r+s,s)$ and $y = (u-2v,3u+v,v)$. Hence, $x + y = (r-2s,3r+s,s) + (u-2v,3u+v,v)$ $= (r-2s+u-2v,3r+s+3u+v,s+v) = ((r+u)-2(s+v),3(r+u)+(s+v),(s+v))$ $= (k-2l,3k+l,l)$, where $k = r + u, l=s+v$. Consequently, $S$ is closed under addition. Further, if $c \in \R$, then $cx = c(r-2s,3r+s,s) = (c(r-2s),c(3r+s),cs) $ $= (cr-2cs,3cr+cs,cs) = (a-2b,3a+b,b)$, where $a = cr,b=cs$. Therefore $S$ is also closed under scalar multiplication. It follows that $S$ is a subspace of $\R^3$.

Problem 107: In a given city, 6% of all drivers get at least one parking ticket per year. Use the Poisson approximation to the binomial distribution to determine the probabilities that among 80 drivers randomly chosen in this given city:
a. 4 will get at least one parking ticket in any given year,
b. At least 3 will get at least at least one parking ticket in any given year,
c. Anywhere from 3 to 6, inclusive will get at least one parking ticket in any given year.

Problem 107: In a given city, 6% of all drivers get at least one parking ticket per year. Use the Poisson approximation to the binomial distribution to determine the probabilities that among 80 drivers randomly chosen in this given city: a. 4 will get at least one parking ticket in any given year, b. At least 3 will get at least at least one parking ticket in any given year, c. Anywhere from 3 to 6, inclusive will get at least one parking ticket in any given year. Solution: Poisson distribution, $f(x;\lambda )=\frac{{\lambda }^{x}{e}^{-\lambda }}{x!}$ a. 4 will get at least one parking ticket in any given year Given that $x=4, n=80, p=0.06, np = 4.8$ $f(4;4.8)=\frac{{4.8}^{4}{e}^{-4.8}}{4!} = 0.182029$ b. At least 3 will get at least at least one parking ticket in any given year Given that $x\ge 3, n=80, p=0.06, np = 4.8$ $P(x\ge 3) = 1-P(x $P(x=0) = f(0;4.8)=\frac{{4.8}^{0}{e}^{-4.8 }}{0!} = 0.00823$ $P(x=1) = f(1;4.8)=\frac{{4.8}^{1}{e}^{-4.8 }}{1!}...

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