GATE Electronics and Communications (EC) 2020 Solved Papers

Question : 17 of 65

Marks: +1, -0

The two sides of a fair coin are labelled as 0 and 1 . The coin is tossed two times independently. Let

M

and

N

denote the labels corresponding to the outcomes of those tosses. For a random variable

{X}

, defined as

{X}=\min ( {M}, {N})

, the expected value

{E}( {X})

(rounded off to two decimal places) is .

Solution:

There can be 4 out comes.

\{ {HH}\},\{ {HT}\},\{ {TH}\},\{ {TH}\}

∴

Let 1 is denoted by head

∴

Let 0 is denoted by Tail.

∴ {M}=\{\table 1 , 1 , 0 , 0\}

{N}=\{\table 1 , 0 , 1 , 0\}

{X}=\min ( {M}, {N})=1000 .

{P}( {X})=\;{1}/{4} \;{1}/{4} \;{1}/{4} 1 {/} 4

Now,

{X}=1

When

\{ {H} {H}\}

comes up

∴ P(X=1)=P[\{H H\}]

=\;{1}/{4}

Now

X=0

when

\{H T\},\{T H\}

\{T T\}

come up When

∴ P(X=0)=\;{1}/{4},

∴ E(x)=\;{1}/{4} × 1=0.25

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