The random variable X has Bernoulli distribution with parameter 𝑝 = 14a) Calculate 𝐸(𝑋) and 𝑉𝑎𝑟(𝑋) (2marks)b) When 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) (1mark)
Question
The random variable X has Bernoulli distribution with parameter 𝑝 = 14a) Calculate 𝐸(𝑋) and 𝑉𝑎𝑟(𝑋) (2marks)b) When 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) (1mark)
Solution
a) The expectation (E) and variance (Var) of a Bernoulli distribution can be calculated as follows:
E(X) = p Var(X) = p(1-p)
Given that p = 1/4,
E(X) = 1/4 Var(X) = 1/4 * (1 - 1/4) = 3/16
b) i) The distribution of Y, the number of successes in n independent Bernoulli trials, is a Binomial distribution.
ii) The variance of a Binomial distribution is np(1-p). Given that the variance of Y is 4, we can set up the equation 4 = n(1/4)(1 - 1/4) and solve for n. However, without knowing the value of n, we cannot determine E(Y). If n was known, E(Y) could be calculated as np.
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