Probability mass function of string of heads from weighted coin,

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Bill Foster

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George Stagg 4 years ago

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Name	Status	Author	Last Modified
Probability mass function of string of heads from weighted coin,	Ready to use	Newcastle University Mathematics and Statistics	05/03/2021 14:32
XProbability mass function of string of heads from weighted coin,	draft	Xiaodan Leng	10/07/2019 22:32

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Question statement

A coin has two faces, $H$ or $T$ . It is weighted so that $P(H)=\var{h}$ and $P(T)=\var{t}$ and is tossed $\var{thismany}$ times.

Let $X$ be the random variable which denotes the longest string of consecutive heads that occur during these tosses.

Compute the:

Probability Mass Function (PMF) of $X$ .
Expectation of $X$ .
Variance of $X$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

h2	number	0.28125
h	number	0.75
h0	number	0.015625
h1	number	0.28125
thismany	integer	3
t	number	0.25
h3	number	0.421875
var	number	0.7536621094
ex	number	2.109375

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Data type

Value

xxxxxxxxxx
 
2*h^2*t

JME function reference

Description

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Generated value: `number`

0.28125

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Ask the student a question, and give any hints about how they should answer this part.

$X$ can take the values $0,\;1,\;2$ or $3$ .

Complete the tables below in order to find the PMF. You must input all numbers as exact decimals - no rounding or approximations.

First, find the probability of each outcome on tossing the coin.

Event	HHH	HHT	HTH	HTT
Probability	Gap 0.	Gap 1.	Gap 2.	Gap 3.
$X$	$3$	$2$	$1$	$1$

Event	THH	THT	TTH	TTT
Probability	Gap 4.	Gap 5.	Gap 6.	Gap 7.
$X$	$2$	$1$	$1$	$0$

Use the information from the above table to find the PMF for $X$ .

$X$	$0$	$1$	$2$	$3$
$P(X=x)$	Gap 8.	Gap 9.	Gap 10.	Gap 11.

Advice

Event	HHH	HHT	HTH	HTT
Probability	$\var{h}^3=\var{h^3}$	$\var{h}^2\times\var{t}=\var{h^2*t}$	$\var{h}^2\times\var{t}=\var{h^2*t}$	$\var{h}\times\var{t}^2=\var{h*t^2}$
$X$	$3$	$2$	$1$	$1$

Event	THH	THT	TTH	TTT
Probability	$\var{h}^2\times\var{t}=\var{h^2*t}$	$\var{h}\times\var{t}^2=\var{h*t^2}$	$\var{h}\times\var{t}^2=\var{h*t^2}$	$\var{t}^3=\var{t^3}$
$X$	$2$	$1$	$1$	$0$

So we see that:

$P(X=0)=\var{h0},\;P(X=1)=\var{h^2*t}+3\times \var{h*t^2}=\var{h1}$

$P(X=2)=2\times \var{h^2*t}=\var{h2},\;P(X=3)=\var{h3}$

The expectation is given by:

$\operatorname{E}[X]=\sum x \;P(X=x)=0\times \var{h0}+1\times \var{h1}+2\times \var{h2}+3\times \var{h3}=\var{ex}$ (exactly).

For this discrete distribution we use the formula:

$\begin{eqnarray*}\operatorname{Var}(X)&=&\operatorname{E}[X^2]-\operatorname{E}(X)^2\\&=&0\times \var{h0}+1\times \var{h1}+4\times \var{h2}+9\times \var{h3}-\var{ex}^2\\&=&\var{precround(var,4)}\end{eqnarray*}$ to 4 decimal places.

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Probability mass function of string of heads from weighted coin,

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George Stagg 4 years ago

Bill Foster 6 years, 1 month ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Probability mass function of string of heads from weighted coin,

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

George Stagg 4 years ago

Bill Foster 6 years, 1 month ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: number

Parts

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Generated value: `number`