Question

Let the random variable X represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of X is .

• A. 1.5

Answer 1

The correct option is A 1.5

Let x be random variable which denotes number of tosses to get two heads.

x	outcome	p(x)
2	HH	${\left(\frac{1}{2}\right)}^2$
3	THH	${\left(\frac{1}{2}\right)}^3$
4	TTHH	${\left(\frac{1}{2}\right)}^4$
⋮	⋮	⋮
⋮	⋮	⋮
⋮	⋮	⋮

$E\left(x\right)=\sum p_i{x}_i$

$=2{\left(\frac{1}{2}\right)}^2+3{\left(\frac{1}{2}\right)}^3+4{\left(\frac{1}{2}\right)}^4+...$

$=\frac{1}{2}[(1+2\left(\frac{1}{2}\right)+3{\left(\frac{1}{2}\right)}^2+4{\left(\frac{1}{2}\right)}^3+.....)-1]$

$=\frac{1}{2}[{(1-\frac{1}{2})}^2-1]$

$(\because (1-x{)}^2=1+2x+3x^2+4x^3+.....)$

$=\frac{1}{2}(4-1)=1.5$