CRV Distribution

Question

I hold a length of 1m string in my hand, with the end trailing to the ground. I am blindfolded and someone cuts the string at a completely random point. Let X be the length of string still in my hand.

What is the distribution of X?

What are the mean and standard deviation?

Probability that X is less than 30cm?

Probability that X is between 45 and 65cm?

Answer 1

Distribution of X: Uniform (0,1)

Mean = 0.5 m or 50 cm

SD ~~.2887 m or 28.87 cm

P(x < .3 m) = 30/ 100 = .3

P(.45 < x < .65) = (65 - 45) / 100 = 20 / 100 = .2

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Explanation:

Let X be the length of the string in your hand. Since the cut is made at a random point along the 1-meter string, the cut location is uniformly distributed. => Uniformly distributed on [0, 1]. - we are using a meter, so it is okay to change in cm.

Mean = (b - a) / 2 = (1 - 0 ) / 2 = .5

SD = (b-a) ^ 2 / 12