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Fudgin [204]
4 years ago
9

PLEASE HELP ME!!!! This is my second question

Mathematics
1 answer:
g100num [7]4 years ago
5 0
 hello : here is a solution 

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Exercise 5.2. Suppose that X has moment generating function
soldi70 [24.7K]

Answer:

a) Mean, E(X) = - 0.5

Variance = = 9.25

b) M_{X}(t)=E(e^{tX})

or

⇒ =\sum e^{tx}p(x)

⇒ =\frac{1}{2}+\frac{1}{3}e^{-4t}+\frac{1}{6}e^{5t}

Step-by-step explanation:

Given:

moment generating function  of X as:

MX(t) = \frac{1}{2} + \frac{1}{3}e^{-4t} + \frac{1}{6} e^{5t}

a)  Now

Mean, E(X) = M_{X}'(t=0)

Thus,

M_{X}'(t)=\frac{1}{3}(-4)e^{-4t}+\frac{1}{6}(5)e^{5t}

or

M_{X}'(t)=\frac{-4}{3}e^{-4t}+\frac{5}{6}e^{5t}

also,

E(X^{2})=M_{X}''(t=0)

Thus,

M_{X}''(t)=\frac{-4}{3}(-4)e^{-4t}+\frac{5}{6}(5)e^{5t}

or

M_{X}''(t)=\frac{16}{3}e^{-4t}+\frac{25}{6}e^{5t}

Therefore,

Mean, E(X) = M_{X}'(t=0)=\frac{-4}{3}e^{-4(0)}+\frac{5}{6}e^{5(0)}

or

Mean, E(X) = - 0.5

and

E(X^{2})=M_{X}''(t=0)=\frac{16}{3}e^{-4(0)}+\frac{25}{6}e^{5(0)}

or

E(X^{2}) = 9.5

also,

Variance(X) = E(X²) - E(X)²

⇒ 9.5 - (-0.5)²

= 9.25

b) Now,

Let f(x) be the PMF of X

Thus,

M_{X}(t)=E(e^{tX})

or

⇒ =\sum e^{tx}p(x)

⇒ =\frac{1}{2}+\frac{1}{3}e^{-4t}+\frac{1}{6}e^{5t}

Therefore,

at x = 0, P(x) = \frac{1}{2}

at x= - 4 ,P(x) = \frac{1}{3}

at x = 5, P(x) = \frac{1}{6}

Thus,

E(X) =\sum xP(x)=0(\frac{1}{2})+(-4)(\frac{1}{3})+5(\frac{1}{6})

or

E(X) = - 0.5

also,E(X^{2})=\sum x^{2}P(x)=0^{2}(\frac{1}{2})+(-4)^{2}(\frac{1}{3})+5^{2}(\frac{1}{6})

E(X^{2})  = 9.5

Hence,

Var(X) = E(X²) - E(X)²

⇒ 9.5 - (-0.5)²

= 9.25

4 0
4 years ago
Work out the area of this circle. Give you answer in terms of pi and state it’s units
tino4ka555 [31]

Answer:

49π

Step-by-step explanation:

area of circle = πr^2

<em>Handy Tip: cherry pies delicious; apple pies are(R) too(2)! so C = πd and A = πr^2</em>

diameter = 14 cm

radius = 14/2 = 7cm

area of circle = π(7)^2

area of circle = 49π

Hope that helps!

3 0
3 years ago
What is the height of the boat in the scale model?<br> What length is the real boat?
Thepotemich [5.8K]

Answer:

15

Step-by-step explanation:

6 0
3 years ago
Ned is riding his bike from his house to the park, both of which are on the same straight road. On his way, he stops at a store,
Anvisha [2.4K]

Answer:

Step-by-step explanationcl:

7 0
4 years ago
A triangle has a side lengths of 3 centimeters, 5 centimeters, and 4 centimeters. Abe used converse of the Pythagorean Theorem t
fgiga [73]
No, Abe is incorrect.

If you don’t know which side is the hypotenuse, you have to try every possibility.

3^ + 4^ = 5^
25 = 25

The triangle is a right triangle. The side which is 5 cm long is the hypotenuse.
8 0
3 years ago
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