For the following exercises, compute the values given within 0.01 by deciding on the appropriate f(x) and a and evaluating L ( x ) = f ( a ) + f ' ( a ) ( x − a ) . Check your answer using a calculator. 56. [T] ( 2.0001 ) 6
For the following exercises, compute the values given within 0.01 by deciding on the appropriate f(x) and a and evaluating L ( x ) = f ( a ) + f ' ( a ) ( x − a ) . Check your answer using a calculator. 56. [T] ( 2.0001 ) 6
For the following exercises, compute the values given within 0.01 by deciding on the appropriate f(x) and a and evaluating
L
(
x
)
=
f
(
a
)
+
f
'
(
a
)
(
x
−
a
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. Check your answer using a calculator.
ball is drawn from one of three urns depending on the outcomeof a roll of a dice. If the dice shows a 1, a ball is drawn from Urn I, whichcontains 2 black balls and 3 white balls. If the dice shows a 2 or 3, a ballis drawn from Urn II, which contains 1 black ball and 3 white balls. Ifthe dice shows a 4, 5, or 6, a ball is drawn from Urn III, which contains1 black ball and 2 white balls. (i) What is the probability to draw a black ball? [7 Marks]Hint. Use the partition rule.(ii) Assume that a black ball is drawn. What is the probabilitythat it came from Urn I? [4 Marks]Total marks 11 Hint. Use Bayes’ rule
Let X be a random variable taking values in (0,∞) with proba-bility density functionfX(u) = 5e^−5u, u > 0.Let Y = X2 Total marks 8 . Find the probability density function of Y .
Let P be the standard normal distribution, i.e., P is the proba-bility measure on R, B(R) given bydP(x) = 1√2πe− x2/2dx.Consider the random variablesfn(x) = (1 + x2) 1/ne^(x^2/n+2) x ∈ R, n ∈ N.Using the dominated convergence theorem, prove that the limitlimn→∞E(fn)exists and find it
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