A fair coin is flipped three times. Let X be the number of heads observed. (a) Give both the possible values and probability mass function of X. (b) Find P(X ≥ 1) and P(X > 1). (c) Compute E[X] and V ar(X).

Answers

Answer 1

Answer:

a)

The possible values are 0, 1, 2 and 3.

The probability mass function of X is:

P(X = 0) = 0.125

P(X = 1) = 0.375

P(X = 2) = 0.375

P(X = 3) = 0.125

b)

[tex]P(X \geq 1) = 0.875[/tex], [tex]P(X > 1) = 0.5[/tex]

c) E(X) = 1.5, Var(X) = 0.75

Step-by-step explanation:

For it time the coin is flipped, there are only two possible outcomes. Either it comes up heads, or it comes up tails. Each toss is independent of other tosses. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The variance of the binomial distribution is:

[tex]V(X) = np(1-p)[/tex]

A fair coin is flipped three times.

This means that [tex]n = 3[/tex]

Fair coin means that it is equally as likely to be heads or tails, so [tex]p = \frac{1}{2} = 0.5[/tex]

a) Give both the possible values and probability mass function of X.

The possible values are from 0 to n, so 0, 1, 2 and 3.

The probability mass function is the probability of each outcome. So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{3,0}.(0.5)^{0}.(0.5)^{3} = 0.125[/tex]

[tex]P(X = 1) = C_{3,1}.(0.5)^{1}.(0.5)^{2} = 0.375[/tex]

[tex]P(X = 2) = C_{3,2}.(0.5)^{2}.(0.5)^{1} = 0.375[/tex]

[tex]P(X = 3) = C_{3,0}.(0.5)^{3}.(0.5)^{0} = 0.125[/tex]

(b) Find P(X ≥ 1) and P(X > 1).

[tex]P(X \geq 1) = P(X = 1) + P(X = 2) + P(X = 3) = 0.375 + 0.375 + 0.125 = 0.875[/tex]

[tex]P(X > 1) = P(X = 2) + P(X = 3) = 0.375 + 0.125 = 0.5[/tex]

(c) Compute E[X] and Var(X).

[tex]E(X) = np = 3*0.5 = 1.5[/tex]

[tex]Var(X) = np(1-p) = 3*0.5*0.5 = 0.75[/tex]


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Answers

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Step-by-step explanation:

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Answers

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Step-by-step explanation:

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Step-by-step explanation:

The screenshot you put is the correct answer, here's another method to get the same answer:

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Answers

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Answers

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Answers

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Step-by-step explanation:

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[tex]\color{plum}\boxed{\tt Reasons : }[/tex]Given. They are vertically opposite angles, so they are equal.The bisector BD divides segment AE into two equal parts which are Segment AC and segment CE.The bisector AE divides the segment BD into two equal parts which are segment BC and segment CD.△ACB≅△ECD under the SAS congruence criterion because two sides and one included angle is equal.

Steps to derive these statements and reasons :

Given :

segment AE and segment BD bisect each other

Which means :

∠BCA and ∠DCE are equal because they are vertically opposite angles. Segment AC = segment CE because the bisector BD divides the segment AE into two equal parts which are segment AC and segment CE.segment BC and segment CD will be equal because bisector AE divides the segment BD into two equal parts which are segment BC and segment CD.

Since two sides and one included angle is equal △ACB≅△ECD under the SAS congruence criterion .

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Answers

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Step-by-step explanation:

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Step-by-step explanation:

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