Question

A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 47.0 and 57.0 minutes. Find the probability that a given class period runs between 50.25 and 51.75 minutes.

Find the probability of selecting a class that runs between 50.25 and 51.75 minutes.

Answer 1

Solution

Back-up Theory

If a continuous random variable, X, is uniformly distributed over the interval (a, b), then the pdf (probability density function) of X is given by

f(x) = 1/(b – a) …………………….………………………………………………………..............................................…(1)

CDF (cumulative distribution function) = P(X ≤ t) = Integral (a to t) of f(x) = (t – a)/(b – a)........................................(2)

From (2), P(X > t) = 1- P(X ≤ t) = (b - t)/(b – a)…………………………............................................………………….(3)

From (2), P(t₁ < X < t₂) = P(X < t₂) - P(X < t₂) = (t₂ – t₁)/(b – a)….........................................……...…………………(4)

Now, to work out the solution,

Let X = length of the class in minutes. Then given, X ~ U(47.0, 57.0)

i.e., vide (1), a = 47 and b = 57.

Probability that a given class period runs between 50.25 and 51.75 minutes

= P(50.25 < X < 51.75)

= (51.75 – 50.25)/(57 – 47) [vide (4)]

= 1.5/10

= 0.15 Answer

Probability of selecting a class that runs between 50.25 and 51.75 minutes is the same as the above probability.

DONE