10 The random variable X denotes the number of hours of...

10 The random variable X denotes the number of hours of cloud cover per day at a weather forecasting centre. The probability density function of X is given by f(x) = {((x-18)^2)/k, 0 <= x <= 24, {0, otherwise, where k is a constant. (i) Show that k = 2016. (ii) On how many days in a year of 365 days can the centre expect to have less than 2 hours of cloud cover? (iii) Find the mean number of hours of cloud cover per day.

Transcript

00:01 Hello students, as per the given question, let us solve the problem step by step, where in the first one we need to find the value of k, where we need to ensure that the probability density function is integrated to 1 over the entire range of the random variable x.

00:17 So, which is from 0 to 24.

00:20 So, it is given as integral 0 to 24 f of x dx is equals to 1.

00:29 So, we need to find the value of k such that the integral of the given function over the range 0 to 24 is equals to 1.

00:37 So, for this we have integral 0 to 24 x minus 18 whole square by k into dx is equals to 1.

00:50 To integrate this we get integral 0 to 24 x square minus 36 x plus 324 by k dx is equals to 1.

01:08 Using the integration formula we have that integral x square dx is equals to 1 by 3 into x cube plus c and integral x dx is equals to 1 by 2 into x square plus c.

01:30 So, where c is the constant of the regression.

01:33 So, we can rewrite the integral equation as 1 by k integral 0 to 24 x square minus 36 x plus 324 dx is equals to 1.

01:57 Now, integrating this it gets 1 by k into 1 by 3 x cube minus 36 1 by 2 x square plus 324 of x which is from 0 to 24 is equals to 1.

02:21 Now, we need to substitute the upper bound minus lower bound we need to do for this equation where we need to substitute the values in the place of x and we need to get the values.

02:31 So, after substituting that that way it gives 1 by k 8352 is equals to 1 where k is equals to 8352.

02:47 So, this is the value that has been obtained.

02:51 So, in this next step let us find the number of days in a year which is less than 2 hours of cloud cover.

02:58 We need to calculate the probability of x which is less than 2.

03:01 Since, x follows the given probability density function we need to use the pdf to calculate the probabilities which is in general given as probability of x is less than 2 is equals to integral 0 to 2 x minus 18 whole square by 8352 dx...

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