Question

. In a sequence of 7 Bernoulli trials with probability of success p, let X be the number of successes not followed immediately by a failure. Find E(X) (you can use indicators)

Answer 1

ANSWER:

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Bernoulli trials means An experiment, or trial, whose outcome can be classified as either a success or failure is
performed.
X = 1 when the outcome is a success
X= 0 when outcome is a failure
If p is the probability of a success then the pmf is,
p(0) =P(X=0) =1-p p(1) =P(X=1) =p
Consider that n independent Bernoulli trials are performed. Each of these trials has
probability p of success and probability (1-p) of failure.

p(0) = P(0 successes in 7 trials) = (1-p)
n {FFFFFFF}
p(1) = P(1 success in 7 trials) = (7 1)p(1-p)
7-1 {FSFFFFF}
p(2) = P(2 successes in 7 trials) = (7 2)p^2
(1-p)
^(7-2) {FSFSFFF}
……
p(k) = P(k successes in 7 trials) = (7 k)p^k
(1-p)^(7-k)
Which is Binomial random variable Bin(n=7,p) with pmf,
p(k) = P(k successes in 7 trials) = (7 k)p^k
(1-p)
^(7-k) for k=0,1,2,….7.

The mean of binomial distribution is given by,

E(X) =n*p= 7p

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