A problem on probability involving binomial distribution. A random sample of n people who walk to work are chosen, what is the probability that at least r of them are injured, given that the probability of being injured while walking to work is p.

moidu13x8 2022-09-15 Answered
A problem on probability involving binomial distribution
I'm trying to figure out the method required to solve this problem, so I stripped out the actual values to keep from getting a direct answer.
A random sample of n people who walk to work are chosen, what is the probability that at least r of them are injured, given that the probability of being injured while walking to work is p.
I don't know where to go. It feels like a binomial probability problem, but it covers a range of trials and not just one value exactly. My guess was to calculate 1 B i n o m C D F ( n , p , r 1 ). Does this seem accurate? For example, if n = 15, r = 7, p = 0.5.
I would have 1 B i n o m C D F ( 15 , 0.5 , 6 ) or
1 i = 1 6 ( 15 i ) 0.5 i ( 1 0.5 ) 15 i .
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Answers (1)

ko1la2h1qc
Answered 2022-09-16 Author has 18 answers
Step 1
You are right, this does use the binomial distribution.
P r ( X = r ) = ( n r ) p r ( 1 p ) n r
That simply gives the probability that "r" events are true from a total of "n" possible events, with the probability of the event happening being "p"
Step 2
So using your values:
i = 7 n P r ( X = i )
= i = 7 n ( 15 i ) 0.5 i ( 1 0.5 ) 15 i

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New questions

Question on designing a state observer for discrete time system
I came through this problem while studying for an exam in control systems:
Consider the following discrete time system
x ( k + 1 ) = A x ( k ) + b u ( k ) , y ( k ) = c x ( k )
where b = ( 0 , 1 ) T , c = ( 1 , 0 ) , A = [ 2 1 0 g ] for some g R
Find a feedback regulation (if there is any) of the form u ( k ) = K x ^ ( k ) where x ^ ( k ) is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders e ( k ) = x ( k ) x ^ ( k ) go to zero after a few finite time. layout the kingdom observer and the block diagram.
My method
it is clean that the eigenvalues of the machine are λ 1 = 2 , λ 2 = g (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle | z | = 1.
The state observer equation is
[ x ( k + 1 ) e ( k + 1 ) ] T = [ A b K B k O A L C ] [ x ( k ) e ( k ) ] T
With characteristic equation
χ ( z ) = | z I A + b K | | z I A + L C | = χ K ( z ) χ L ( z )
Also consider
K = [ k 1 k 2 k 3 k 4 ]
and let a = k 1 + k 3 , β = k 2 + k 4
Then χ K ( z ) = ( z 2 ) ( z + g + β ) + a.
So we can select some eigenvalues inside the unit circle and determine a , β in terms of g. Choosing e.g. λ 1 , 2 = ± 1 / 2 we get a = 3 g + 33 / 8 , β = 9 / 4 g , g R
Questions
I want to ask the following:
Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose | λ 1 , 2 | < 1?
Should I determine L matrix as well since the error must also vanish? (because it is not asked)