Question

(). (a) Find cov(W, X) fr W and % delin(xl in I'roblemi l (b) The joint density of random variables X and Y is 0, elsewhere.

Answer 1

$(1)~(a)~P(W=0,Z=0)=\frac{\binom{48}{2}}{\binom{52}{2}}=\frac{188}{221}\\ Since~there~4~aces~and~\{W=0,Z=0\}~means~in~two~draws~there~are~no~ace.\\ P(W=1,Z=0)=\frac{\binom{48}{1}\binom{4}{1}}{\binom{52}{2}}=\frac{32}{221}\\ Since~there~4~aces~and~\{W=1,Z=0\}~means~in~two~draws~first~draw~shows~ace~and\\ second~draw~does~not~show~ace.\\ P(W=1,Z=1)=\frac{\binom{4}{2}}{\binom{52}{2}}=\frac{1}{221}\\ Since~there~4~aces~and~\{W=1,Z=1\}~means~in~two~draws~~aces~are~appeared.\\ P(W=w,Z=z)=0~otherwise\\ (b)~W~and~Z~are~not~independent~since\\ P(W=1,Z=1)\neq P(W=1)P(Z=1)\\ where,~P(W=1)=P(W=1,Z=0)+P(W=1,Z=1)=\frac{33}{221}\\ P(Z=1)=P(W=1,Z=1)=1/221\\ (5)(a)~E(WZ)=P(W=1,Z=1)=\frac{1}{221}\\ P(W=0)=\frac{188}{221},~P(W=1)=P(W=1,Z=0)+P(W=1,Z=1)=\frac{33}{221}\\ E(W)=P(W=1)=\frac{33}{221}\\ E(Z)=P(Z=1)=P(W=1,Z=1)=\frac{1}{221}\\ Cov(W,Z)=E(WZ)-E(W)E(Z)=1/221(1-33/221)\\ =188/(221)^2=0.0038$