A random variable N has a cumulative distribution function given by

F(x)={ 0 for x<1
0.2 for 1<=x<2
0.5 for 2<=x<3
0.8 for 3<=x<4
1 for 4<=x
Find the mean of N.

To find the mean of a random variable, you need to use the cumulative distribution function (CDF) and the individual probabilities associated with each interval. First, let's identify the intervals for which we have defined probabilities.

From the given CDF, we have:

P(N < 1) = 0
P(1 ≤ N < 2) = 0.2
P(2 ≤ N < 3) = 0.5
P(3 ≤ N < 4) = 0.8
P(N ≥ 4) = 1

Now, we can calculate the mean by using the following formula:

Mean (μ) = ∑ (x * P(x))

In this formula, x represents the observed value of the random variable, and P(x) represents the probability associated with that value.

Let's calculate the mean step by step:

For the interval (1 ≤ N < 2):
Mean = (1 + 2) * (0.2 - 0) / 2 = 0.6 / 2 = 0.3

For the interval (2 ≤ N < 3):
Mean = (2 + 3) * (0.5 - 0.2) / 2 = 2.5 * 0.3 = 0.75

For the interval (3 ≤ N < 4):
Mean = (3 + 4) * (0.8 - 0.5) / 2 = 3.5 * 0.3 = 1.05

For the interval (N ≥ 4), we assume the value of N as 4:
Mean = 4 * (1 - 0.8) = 4 * 0.2 = 0.8

Now, we can sum up all the means calculated for each interval to get the overall mean:

Mean (μ) = 0.3 + 0.75 + 1.05 + 0.8 = 2.9

Therefore, the mean of the random variable N is 2.9.