Answer to Question #309911 in Statistics and Probability for ced

Question

2. A fire-detection device uses three-temperature-sensitive cells acting independently of

one another in such a manner that any one or more can activate the alarm. Each cell has

a probability p = 0.8 of activating the alarm when the temperature reaches 100° or

higher. Let X represent the number of cells activating the alarm when the temperature

reaches 100°. Find:

a) the probability distribution of X;

b) the expected value; and

c) the variance for the random variable X.

Answer 1

a)

Let X represent the number of cells activating the alarm when the temperatures reaches 100°. Also, let Y represent the number of cells not activating the alarm when the temperature reaches 100

Then, the sample spaces are:

X=0; {Y,Y,Y}

X=1;{X,Y,Y}{Y,X,Y}{Y,Y,X}

X=2; {X,X,Y}{X,Y,X}{Y,X,X}

X=3; {X,X,X}

The probability distribution of X is given by:

X. P(X)

0. 1/8

1. 3/8

2. 3/8

3. 1/8

b)

The expected value E(X) = ∑X*P(X)

=0*1/8 + 1*3/8 +2*3/8 + 3*1/8

= 1.5

c)

Var(X) = ∑P(X)*(X- E(X))²

= 1/8*(0-1.5)²+3/8*(1-1.5)²+3/8*(2-1.5)²+1/8*(3-1.5)²

=0.75