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Question
Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If P(x = r)/P(x = n – r) is independent of n and r, then p equals ______.
Options
`1/2`
`1/3`
`1/5`
`1/7`
Solution
Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If P(x = r)/P(x = n – r) is independent of n and r, then p equals `1/2`.
Explanation:
P(X = r) = `""^"n""C"_"r" "p"^"r" "q"^("n" - "r") = ("n"1)/("r"!("n" - "r")!) "p"^"r"*(1 - "p")^("n" - "r")`
P(X = n – r) = `""^"n""C"_("n" - "r") "p"^("n" - "r") * ("q")^("n" - ("n" - "r")`
= `""^"n""C"_("n" - "r") "p"^("n" - "r") * "q"^"r"`
= `("n"1)/(("n" - "r")!("n" - "n" + "r")!) "p"^("n" - "r")"q"^"r"`
= `("n"!)/(("n" - "r")!"r"!) * "p"^("n" - "r") * "q"^"r"`
Now `("P"(x = "r"))/("P"(x = "n" - "r"))`
= `(("n"!)/("r"!("n" - "r")!)*"p"^"r"*(1 - "p")^("n" - "r"))/(("n"!)/("r"!("n" - "r")!)*"p"^("n" - "r")*(1 - "p")^"r")`
= `((1 - "p")/"p")^("n" - "r")/((1 - "p")/"p")^"r"`
The above expression will be independent of n and r if
`((1 - "p")/"p")` = 1
⇒ `1/"p"` = 2
⇒ p = `1/2`
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