Probability Distribution - Distribution Function of a Continuous Random Variable

If the p.d.f. of a continuous random variable X is

f(x) = k (x - x²),    0 < x < 1

= 0,             otherwise

then the value of k is ______.

If the p.d.f of a c.r.v. X is
f(x) = `{:("K.e"^(-thetax)";", theta > 0","  0 ≤ x < oo),(0";", "otherwise"):}`
Then, K = ______.

If the p.m.f of a. r.v. Xis given by

Givenif x = 0, 1, 2, ........ 5 = 0,

0, otherwise,

then which of the following is not true?

The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class.
Also prepare marginal distributions :

(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )

A random variable X has the following has the following:

Then, the value of P(–1 < X < 3) is ______.

Compute correlation coefficient for the following data :

If a random variable X follows Poisson distribution sucli that P(X = 1) = P(X = 2), find:
(a) the mean and
(b) P(X = 0). [Given: e^-2 = 0.1353)

Solve the inequation -8 ≤ 3x - 5 < 7 and represent it on the real number line.