hen n independent Bernoulli trials are carried out with 𝑝 = 14, the number of successes,Y, can be modelled by another distribution.i) State the distribution of Y. (1mark)ii) Given that the variance of Y is 4, determine 𝐸(𝑌)
Question
hen n independent Bernoulli trials are carried out with 𝑝 = 14, the number of successes,Y, can be modelled by another distribution.i) State the distribution of Y. (1mark)ii) Given that the variance of Y is 4, determine 𝐸(𝑌)
Solution
i) The distribution of Y, the number of successes in n independent Bernoulli trials with probability of success p = 1/4, is a Binomial distribution. This is because a Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials.
ii) The variance of a Binomial distribution is given by np(1-p). Given that the variance of Y is 4, we can set up the equation 4 = n(1/4)(1 - 1/4) = n(1/4)(3/4) = 3n/16. Solving for n gives n = 64/3.
The expected value E(Y) of a Binomial distribution is np. Substituting n = 64/3 and p = 1/4 into this formula gives E(Y) = (64/3)(1/4) = 16/3.
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