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प्रश्न
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.
उत्तर
Let X denote the number of heads in the four tosses of the coin.
Then X is a random variable that can take the values 0, 1, 2, 3 or 4.
P(X=0)=Probability of getting no head (TTTT)
`=1/16`
P(X=1)=Probability of getting one head (HTTT, THTT, TTHT, TTTH)
`=4×1/16=1/4`
P(X=2)=Probability of getting two heads (HHTT, HTHT, HTTH, THHT, THTH, TTHH)
`=6×1/16=3/8`
P(X=3)=Probability of getting three heads (HHHT, HHTH, HTHH, THHH)
`=4×1/16=1/4`
P(X=4)=Probability of getting four heads (HHHH)
`=1/16`
The probability distribution of X is
| X | 0 | 1 | 2 | 3 | 4 |
| P(X) | 1/16 | 1/4 | 3/8 | 1/4 | 1/16 |
| xi | pi | pixi | pixi2 |
| 0 | 1/16 | 0 | 0 |
| 1 | 1/4 | 1/4 | 1/4 |
| 2 | 3/8 | 3/4 | 3/2 |
| 3 | 1/4 | 3/4 | 9/4 |
| 4 | 1/16 | 1/4 | 1 |
| ∑pixi=2 | ∑pixi2=5 |
Mean, E(X) = ∑pixi=2
Var(X)= ∑pixi2−(∑pixi)2=5−4=1
Therefore, the mean and variance of the number of heads obtained are 2 and 1, respectively.
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