Problem 73: Prove that the subset $W = \{(x,2x,3x)|x \in \R\}$ of $\R^3$ is a subspace of $\R^3$.

Solution:
$W = \{(x,2x,3x)|x \in \R\}$.
zero check:
If we take $x = 0$, we see that $(0,0,0) \in W$ , so W is non-empty.
Now let $u = (x,2x,3x)$ and $v = (y,2y,3y)$ be any two elements of W for $x, y \in \R$.Then
Closure under Addition:
$u+v = (x,2x,3x) + (y,2y,3y) = (x+y,2x+2y,3x+3y)$
$((x+y),2(x+y),3(x+y)) = (z,2z,3z) \in W$
if $z=x+y$
So, $u+v \in W$
Hence,W is closed under addition.
Closure under Scalar Multiplication:
Let $k \in \R$ and $u=(x,2x,3x) \in W$
Then, $ku = (kx,2kx,3kx)$
which shows that $ku \in W$
Therefore W is a subspace of $\R^3$

Comments

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) =...

upāyaṁ karoti

Search This Blog