Question

# 1.if a is uniformly distributed over [−12,24], what is the probability that the roots of the...

1.if a is uniformly distributed over [−12,24], what is the probability that the roots of the equation x2+ax+a+24=0 are both real?

Hint: The roots are real if the discriminant in the quadratic formula is positive.

2.The time (in minutes) between arrivals of customers to a post office is to be modelled by the Exponential distribution with mean .38. Please give your answers to two decimal places.

Part a)

What is the probability that the time between consecutive customers is less than 15 seconds?

Part b)

Find the probability that the time between consecutive customers is between ten and fifteen seconds.

Part c)

Given that the time between consecutive customers arriving is greater than ten seconds, what is the chance that it is greater than fifteen seconds?

SOLUTION:

From given data,

1.if a is uniformly distributed over [−12,24], what is the probability that the roots of the equation x2+ax+a+24=0 are both real?

Hint: The roots are real if the discriminant in the quadratic formula is positive.

x2+ax+(a+24)=0

Determinant>0

a2 -4 * (a+24)>0

a2 -4a -96 >0

a2 - 12a + 8a - 96 >0

a(a-12)+8(a-12) >0

(a-12)*(a+8) >0

Hence

a<-8 or a>12

Therefore probability that the roots are real

Hence required probability

f(X) = 1/(24 - (-12))

= 1/36

F(X) = (X -(-12)) / 36

= X+12 /36

= [24-12]/36 + [-8-(-12)]/36

=( 12 / 36) + (4/36)

= 16 /36

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