Let 𝑋∼Bernoulli(0.4)X∼Bernoulli(0.4) and 𝑌∼Bernoulli(0.8)Y∼Bernoulli(0.8) be independent. Define 𝑍=𝑋+𝑌−𝑋𝑌Z=X+Y−XY, find the distribution of 𝑍Z.Bernoulli(0.88)Bernoulli(0.88)Bernoulli(0.12)Bernoulli(0.12)Bernoulli(0.92)Bernoulli(0.92)Bernoulli(0.08)Bernoulli(0.08)

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)

Let 𝑋X and 𝑌Y be independent and identically distributed Geometric random variables with parameter 0.80.8.Find 𝑃(𝑋=4∣𝑋+𝑌=8).P(X=4∣X+Y=8).Enter the answer correct to two decimal places.

The joint pdf of two continuous random variables 𝑋X and 𝑌Y is given by𝑓𝑋𝑌(𝑥,𝑦)={4𝑥𝑦1440≤𝑥≤14,0≤𝑦≤140otherwisef XY​ (x,y)={ 14 4 4xy​ 0​ 0≤x≤14,0≤y≤14otherwise​ Are 𝑋X and 𝑌Y independent?YesNo

The joint distribution of 𝑋X and 𝑌Y is given by                             𝑓𝑋𝑌(𝑥,𝑦)=916×4𝑥+𝑦,f XY​ (x,y)= 16×4 x+y 9​ ,where 𝑇𝑋,𝑇𝑌∈{0,1,2,…}.T X​ ,T Y​ ∈{0,1,2,…}.1 pointFind the probability mass function of 𝑋+𝑌X+Y.𝑘916⋅4𝑘k 16⋅4 k 9​ (𝑘+1)916⋅4𝑘(k+1) 16⋅4 k 9​ (𝑘+1)916⋅4𝑘+1(k+1) 16⋅4 k+1 9​ 𝑘916⋅4𝑘+1k 16⋅4 k+1 9​

Let the discrete RV 𝑋~𝑈[−2,2] (Uniform dist.). Let 𝑌 = 𝑋2a) What values X and Y can take? Find pdf’s of both X and Y.b) Compute the joint pdf, 𝑓𝑋𝑌(𝑥𝑖, 𝑦𝑖)c) Compute the E(X) and E(Y)d) Compute the Cov(X,Y)e) Compute the 𝜌𝑋𝑌 = 𝐶𝑜𝑟(𝑋, 𝑌).f) Are X and Y independent? Prove it.