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MathematicsProbabilityRandom Variable and its Probability DistributionMedium2 minPYQ_2020

MathematicsMediumsingle choice

An unbiased coin is tossed $5$ times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k = 3, 4, 5,$ otherwise $X$ takes the value $- 1 .$ Then the expected value of $X,$ is

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Answer:

Solution:

If $k$ represents the number of consecutive heads and $P (k)$ is the corresponding probability while an unbiased coin is tossed $5$ times, then

$k$	$0$	$1$	$2$	$3$	$4$	$5$
$P (k)$	$\frac{1}{2^{5}}$	$\frac{12}{2^{5}}$	$\frac{11}{2^{5}}$	$\frac{5}{2^{5}}$	$\frac{2}{2^{5}}$	$\frac{1}{2^{5}}$

Now, the expected value of variable $X$ is

$(- 1) \frac{1}{2^{5}} + (- 1) \frac{12}{2^{5}} + (- 1) \frac{11}{2^{5}} + (3) \frac{5}{2^{5}} + (4) \frac{2}{2^{5}} + (5) \frac{1}{2^{5}}$

$= \frac{1}{2^{5}} [- 1 - 12 - 11 + 15 + 8 + 5]$

$= \frac{28 - 24}{2^{5}} = \frac{4}{2^{5}} = \frac{1}{2^{3}} = \frac{1}{8}$

Stream:JEESubject:MathematicsTopic:ProbabilitySubtopic:Random Variable and its Probability Distribution

⏱ 2mℹ️ Source: PYQ_2020

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