Question

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Question # 3 (A) The probability distribution function of a continuous variable is as follows: p(x) k x (1+x2) 0 G1Sx3+1) else where where k is a constant. (i) (ii) (iii) Find the value of, k Find the expected mean and standard deviation Determine the cumulative probability between (-1 that is, P(x+0.5)? x + 0.5); Answer-hints: (iii)i P(x 0.5) 15/16] Indicated below is a data set of eight numbers depicting random occurrences of an epoch x. The assigned numbers (1. 2,. 8) for the RV, x represent respectively, the directions (N, S, E, W, NE, NW, SE and SW) of a hurricane-track; and, the corresponding occurrence probabilities of x are: p(x) as listed. Data set B-I (B-1) 4 p(x)0.10 0.20 0.18 0.15 0.05 0.05 0.12 0.15 (i) Determine the expected mean of x (ii) Find the standard deviation of the distribution of x (ii) Decide the most probable direction of the hurricane based on the central-tendency Performance of the underlying statistics. (vi) Plot: CDF of x

Answer 1

(A)(i)

For a valid pdf sum of all probabilities must be equal to 1. Now,

So,

(ii)

The expected value is

Now,

The variance of X is

(iii)

(B-1)

Following table shows the calculations:

(i)

The expected value is :

(ii)

The standard deviation is

(iii)

Since expected value is 4.23 so most probable direction is W.

(vi)

Following table shows the CDF:

Following is the graph: