Random Variables and Its Probability Distributions

Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}`
where k is a constant. Calculate Standard deviation of X.

Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}`
where k is a constant. Calculate the value of k

10 balls are marked with digits 0 to 9. If four balls are selected with replacement. What is the probability that none is marked 0?

Let X represent the difference between the number of heads and the number of tails when a coin is tossed 6 times. What are the possible values of X?

Two fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. Also find E(X).

Solve the following :

Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

20 white rats are available for an experiment. Twelve rats are male. Scientist randomly selects 5 rats number of female rats selected on a specific day

Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that

where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.

Find the probability of throwing at most 2 sixes in 6 throws of a single die.