A newly developed software is used to detect potential threats in two different systems. The software is able to detect threats with probability 0.6 in the first system. Independently of it, threats in the second system is detected with probability 0.5

a.Compute the probability mass function (pmf) of X, the number of systems in which their threats were detected.

b.Draw a graph of its cumulative distribution function (cdf).

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21-02-2020
Mathematics

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A newly developed software is used to detect potential threats in two different systems. The software is able to detect threats with probability 0.6 in the first system. Independently of it, threats in the second system is detected with probability 0.5

a.Compute the probability mass function (pmf) of X, the number of systems in which their threats were detected.

b.Draw a graph of its cumulative distribution function (cdf).

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salwanadeem salwanadeem · Answer 1 · 2020-02-23T13:44:12+01:00

Answer:

a) x 0 1 2

P(X=x) 0.2 0.5 0.3

b) CDF attached as an image

Step-by-step explanation:

a) Let us denote probability of threat in system 1 with P(T1) and probability of threat in system 2 with P(T2).

P(T1) = 0.6

P(T2) = 0.5

The probability that there is no threat in system 1 is:

P(T1)' = 1 - P(T1)

= 1 - 0.6

P(T1)' = 0.4

Similarly,

P(T2)' = 1 - P(T2)

= 1 - 0.5

P(T2)' = 0.5

The sample points for this situation can be:

(T1,T2) (T1',T2) (T1,T2') (T1',T2')

X = number of systems in which threats were detected. i.e. 0,1,2

P(X=0) = P(T1',T2')

= P(T1)' * P(T2)'

= 0.4 * 0.5

P(X=0) = 0.2

P(X=1) = P(T1,T2') + P(T1',T2)

= P(T1) * P(T2)' + P(T1)' * P(T2)

= 0.6*0.5 + 0.4*0.5

=0.3 + 0.2

P(X=1) = 0.5

P(X=2) = P(T1,T2)

= P(T1) * P(T2)

= 0.6*0.5

P(X=2) = 0.3

PMF of X:

x 0 1 2

P(X=x) 0.2 0.5 0.3

b) The cumulative distribution function can be plotted by adding up the probabilities of each point with that of its previous points. i.e.

P(X≤0) = P(X=0) = 0.2

P(X≤1) = P(X=0) + P(X=1) = 0.2 + 0.5 = 0.7

P(X≤2) = P(X=0) + P(X=1) + P(X=2)

= 0.2 + 0.5 + 0.3

P(X≤2) = 1

Now we need to plot this CDF. On the x-axis, we will plot the values of X and on the y-axis, we will plot the values of CDF. I am attaching the plot as an image.