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A secretary makes 2 errors per page on the average. What is the probability that on the next
page she makes; 4 or more errors and no error?
Let X be a random variable with probability distribution
X -1 0 1 2 3
F(X) 0.125 .50 0.20 0.05 0.125
a) Find E(X) and VAR(X).
b) Find the probability distribution of the random variable Y= 2X+1. Using the
probability distribution of Y determine E(Y) and VAR(Y).
A population consists of the values (1, 3, 4). From the list below, choose 9 possible samples of size 2 that can be drawn from this population with replacement
Check whether the sequence (an), where
an = 1/ (n+1) + 1/(n+2) +....+1/(2n) is convergent or not
Let ϕ and Ψ be function defined on [-3,5], such that both are continuous on [-3,5], derivable in [-3,5] and ϕ'(x)= Ψ'(x) ∀ x∈]-3,5[. Prove that
ϕ(x)= Ψ(x) +c ∀ x∈ [-3,5] , where c is a real constant
Let f:[ 0,π/2] → [-1,1] be a function defined by f(x)= cos 2x . Verify that f satisfies the condition of the inverse function theorem. Hence, what can you conclude about the continuity of f^-1?
Prove that the sequence (fn(x)), where fn(x)= nx/(1+ nx^2) is not uniformly convergent in [-2,2]
Check whether or not the function f, defined on R by
f(x) = { 3x^2sin(1/2x), when x≠0
{ 0 ,when x=0
is derivable on R. If it is, is f' continuous at x=0? If f is not derivable , then define a derivable function on R
Let the function , defined by
f(x)= 1/x+3 ; x∈ [3,∞[
Check whether f is uniformly continuous or not on the interval of definition
1. Show that if 5/3< 2x< 11/3, then x∈{y∈R such that |y- 4/3| < 1/2}
2. Draw the graph of the function f, defined by f(x)= |x-6| + |5 - x| ; x∈ [2,8]
