[Probability and Statistics] I\'m really stuck in this problem, any help would b

Question

[Probability and Statistics] I'm really stuck in this problem, any help would be really appreciated, Thanks!

Suppose that a non-negative random variable X is distributed according to a probability mass function f_x. We know the expectation of the random variable is E[X] = 1. a. Provide an upper bound on the event P(X greaterthanorequalto 8). b. Suppose that the p.m.f. f_x is given by f(x) = {7/8 x = 0 1/8 x = 8 0 else. Confirm that f is a p.m.f., that the expectation of a random variable with this distribution is equal to 1, and compute P(X greaterthanorequalto 8).

Explanation / Answer

as we know from Markov's Inequality P(X>=t)<=E(X)/t

a) therefore upper bound on event P(X>=8) is E(X)/8 =1/8

b)

for f needs to be a pmf ; the sum of its probability over its domain should be equal to 1 and individual probability should be non negative and less then 1.

here as f(0)+f(8) =7/8+1/8 =1 ; therefore f is a pmf

expectation of X =E(X) =0*(7/8)+8*(1/8)=1

P(X>=8) =1/8 from above pmf.

please revert for any clarificaiton required

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