Burglar Alarm Components
statistic question?
a burglar Alarm System has six fail-safe components. The probability of each failing is o.oo5. Find these probabilities: Exactly 3 will fail, fewer than 2 will fail and none will fail. Then compare the answers and explain why the results are reasonable. I got the number for the first 3 question, but i still confused about the last one. Explain why the result are reasonable??? Can you give me your ideas?
Let X be the number of components that fail. X has the binomial distribution with n = 6 trials and success probability p = 0.005
In general, if X has the binomial distribution with n trials and a success probability of p then
P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)
for values of x = 0, 1, 2, …, n
P[X = x] = 0 for any other value of x.
The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n – x failures.
Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.
X ~ Binomial( n = 6 , p = 0.005 )
the mean of the binomial distribution is n * p = 0.03
the variance of the binomial distribution is n * p * (1 – p) = 0.02985
the standard deviation is the square root of the variance = √ ( n * p * (1 – p)) = 0.1727715
The Probability Mass Function, PMF,
f(X) = P(X = x) is:
P( X = 0 ) = 0.9703725 ← none will fail
P( X = 1 ) = 0.02925746
P( X = 2 ) = 0.0003675561
P( X = 3 ) = 2.462687e-06 ← exactly three will fail
P( X = 4 ) = 9.281484e-09
P( X = 5 ) = 1.865625e-11
P( X = 6 ) = 1.5625e-14
The Cumulative Distribution Function, CDF,
F(X) = P(X ≤ x) is:
x
∑ P(X = t) =
t = 0
P( X ≤ 0 ) = 0.9703725
P( X ≤ 1 ) = 0.99963 ← fewer than 2 will fail
P( X ≤ 2 ) = 0.9999975
P( X ≤ 3 ) ≈ 1
P( X ≤ 4 ) ≈ 1
P( X ≤ 5 ) ≈ 1
P( X ≤ 6 ) = 1
1 – F(X) is:
n
∑ P(X = t) =
t = x
P( X ≥ 0 ) = 1
P( X ≥ 1 ) = 0.02962749
P( X ≥ 2 ) = 0.0003700281
P( X ≥ 3 ) = 2.471987e-06
P( X ≥ 4 ) = 9.300156e-09
P( X ≥ 5 ) = 1.867184e-11
P( X ≥ 6 ) = 1.554312e-14
Installing Your Burglar Alarm
The complexity of an alarm system depends on the establishment it is going to be used. The more entry points, the more complicated the electrical circuit for The Alarm system. Most of the time, complex alarm systems are installed in business establishments like stores. The need for a complex alarm system for this kind of establishment is higher than any building.
Most homes use simpler versions of alarm systems. In many cases, smaller homes use simple unmonitored alarm. Using alarm systems in homes is grounded in the concept that even homes and the persons residing in it needs protection too.
Alarm systems vary according to the task it is going to do. For example, when a person wanted to detect smoke or fire, he may choose installing Fire Alarm system in his house. Every type of alarm system works specifically to a certain job. The most common used alarm system type is the burglar alarm system.
Burglar Alarm Systems vary too according to the size of the building. Also, the installation complexity of such may change depending on the number of entry points to be integrated. In addition, Burglar Alarms may also be monitored by the police or relayed to a telephone line.
Most burglar alarm systems are installed by professionals. In many cases, the company where one bought the alarm systems is the one responsible for the installation of the alarm system.
However, there are also alarm systems that can be installed by do-it-yourselfers. These alarm systems are being sold in kits. Every kit contains complete components that one needed for the installation.
Before installing an alarm systems take note of the following considerations for a proper installation:
• The number of entry points in the house. Knowing how many entry points you have in your home will determine how many sensors you need to install. Entry points are not limited to doors. It can also be windows, fire exits, etc.
• The type of alarm system you brought. Is it a burglar alarm or a combination of a burglar alarm and a fire alarm? The more sensor types the complex the circuit will be.
• Note of the strategic place where you are going to put the control panel.
In installing the burglar alarm system, the very first component to be mounted must be the sounder or other known as the buzzer or bell. Think of where you want it to put. Most of the time, sounders are located in places that can easily be seen to warn off any burglar intending to break into the house. When the sounder is installed outside, one must make sure that it is mounted away from extreme heat or moisture. It is also advisable to enclose the sounder in a box to protect it from any atmospheric condition. Make sure also, that the sounder is placed in a way that it can be heard loudly.
After installing the sounder, the next component that must be installed is the sensors or the switches. Carefully install the sensors in edges of the windows and doors. Make sure that the magnet and the sensor is not too far from each other so that the sensor will have enough magnet reception. Do not put it otherwise too close to each other or else the magnet may sag to the sensor which can damage any of the component part.
Mount the wires neatly and integrate the rest of the components to the control panel. Test the system if it works properly by closing and opening the entry points. Make adjustments when there are components that do not work properly.
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