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Exercise 3.62. A little boy plays outside in the yard. On his own he would come back inside at a random time uniformly distributed on the interval [0, 1]. (Let us take the units to be hours.) However, if the boy is not back inside in 45 minutes his mother brings him in. Let X be the time when the boy comes back inside (a) Find the cumulative distribution function F of X (b) Find the mean E(X) (c) Find the variance Var(X) Hint. You should see something analogous in Examples 3.20 and 3.38.

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Answer:

0.38

−0.0814

Step-by-step explanation:

X ___ 0 __ 0.15 __ 0.30 ___ 0.45 ___ 1

P(x)_0.20 _0.20___0.20 ___0.25 __0.20

The expected mean ; E(X) :

Σx * p(x)

(0*0.2) + (0.15*0.2) + (0.3*0.2) + (0.45*0.2) + (1*0.2) = 0.38

The variance ; Var(X) :

Σx²*p(x) - E(x)²

(0^2 * 0.2) + (0.15^2 * 0.2) + (0.3^2 * 0.2) + (0.45^2 * 0.2) + (1*0.2) - 0.38^2

0.063 - 0.1444

−0.0814

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