Let X be a positive random variable (i.e. P(X <0) = 0. Argue that (a) E(1/X) > 1/E(X) (b) E(-log(X)) > -log(E(X)) (c) E(log(1/X)) > log(1/E(X)) (d) E(X³) > (E(X))³

Q: Estimate g(1.2) − g(1) if g'(1) = 4.

A: Given: g'(1)=4 Now, Using the Linear approximation formula:

Q: Suppose that the time in minutes, Y , that an insect spends on a plant is exponentially distributed…

A: Y is the time in minutes that an insect spends on a plant and .

Q: Differentiate f(x) = log, (9 – 10x).

A:

Q: 2. Estimate g(1.2) – g(1) if g'(1) = 4. -

A: Given, g'(1)=4. The formula for the linear approximation is, g(x)=g(a)-g'(a)(x-a), where x is near…

Q: log, (4) = 1 %3D

A:

Q: Prove that log(x + y) = log(x) + log(y) if, and only if, y = 1, where x > 1.

A:

Q: (b) 8[log(2-j3)]e

A:

Q: 3. Differentiate. x²e* f(x)%3= x² + ex

A:

Q: Let f(x) be piecewise continuous and a -exponential. In this case, prove that F(s) L £{f(x)} = F(s)…

A:

Q: If Y Exponential(8), a > 0 and b > 0, show that P(Y > a+b[Y > a) = e-b/B.

A: P[A|B] = P[A∩B]/P[B] So, P[Y&gt;a+b | Y&gt;a] = P[Y&gt;a+b ∩ Y&gt;a] / P[Y&gt;a] for a&gt;0, b&gt;0,…

Q: Suppose f(x) = sinx and g(x) = logx. %3D %3D a) State the equation of the function y = g[f(x)] %3D…

A:

Q: a) f (x) = 7x log 3x b) f (x) = 3*

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.