Suppose X is a discrete random variable taking values 1, 2, . . ., 10, with probability function fX(), and cumulative distribution function FX(). Say whether each of the following statements is true or false.

a. The values FX(x) must sum to 1 over x = 1, 2, . . . , 10.

b. The values of FX(x) must all be between 0 and 1 inclusive.

c. Each value FX(x) is a probability.

d. The values fX(x) must sum to 1 over x = 1, 2, . . . , 10.

e. Each value fX(x) is a probability.

f. fX(2.5) = 0, because X cannot take the value 2.5.

g. FX(2.5) = 0, because X cannot take the value 2.5.

h. FX(10.5) = 1.

i. fX(10.5) = 1.28.4 Suppose X is a discrete random variable taking values 1, 2, . . ., 10, with probability function fX(), and cumulative distribution function FX(). Say whether each of the following statements is true or false.

a. The values FX(x) must sum to 1 over x = 1, 2, . . . , 10.

b. The values of FX(x) must all be between 0 and 1 inclusive.

c. Each value FX(x) is a probability.

d. The values fX(x) must sum to 1 over x = 1, 2, . . . , 10.

e. Each value fX(x) is a probability.

f. fX(2.5) = 0, because X cannot take the value 2.5.

g. FX(2.5) = 0, because X cannot take the value 2.5.

h. FX(10.5) = 1.

i. fX(10.5) = 1.a. Write down the cumulative distribution function (CDF), FX(x), in a similar tabular format to fX(x) above.

b. Using the CDF, find the following probabilities:
i. P(X ≤ 4).
ii. P(X > 6).
iii. P(2 ≤ X ≤ 5).

c. Benford’s law is remarkable because a first digit of 1 alone has the same chance as any of the digits from K to 9 inclusive. Find which number K this statement refers to.
Start by writing out a probability expression that K needs to satisfy, then do the necessary calculations to find K.
Finish your answer with a sentence in plain English to summarise your result.