Suppose the proportion X of surface area in a randomly selected that is covered

Question

Suppose the proportion X of surface area in a randomly selected that is covered by a certain plant has a standard beta distribution with alpha = 4 and beta = 2. Compute E(X) and V(X). (Round your answers to four decimal places.) E(X) = 0 6667 V(X) = 0.0317 Compute P(X lessthanorequalto 0.2). (Round your answer to four decimal places.) Compute P(0.2 lessthanorequalto X lessthanorequalto 0.8). (Round your answer to four decimal places.) What is the expected proportion of the sampling region not covered by the plant? (Round your answer to four decimal places.)

Explanation / Answer

x follows standard Bets distribution with parameter alpha = 4 and beta = 2

rang of x is ( 0, 1 )

b ) P( x <= 0.2 )

Using Excel,

=BETADIST( X, alpha, beta, lower bound of x , upper bound of x )

=BETADIST( 0.2 , 4, 2, 0, 1 ) = 0.0067

c) P( 0.2 <= x <= 0.8 ) = p( x <= 0.8 ) - P( x <= 0.2 )

Using Excel,

P( x <= 0.8 ) = BETADIST( 0.8, 4, 2, 0, 1 ) = 0.7373

p( x <= 0.2 ) = 0.0067

P( 0.2 <= x <= 0.8 ) = 0.7373 - 0.0067 = 0.7306

d) Expected proportion of the sampling region covered by the plant, E(x) = 0.6667

Expected proportion of the sampling region not covered by the plant is,

= 1 - E(x) = 1 - 0.6667

= 0.3333

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