Advertisements
Advertisements
प्रश्न
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 ______.
पर्याय
`1/2`
`1/3`
`1/5`
`1/7`
उत्तर
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`
APPEARS IN
संबंधित प्रश्न
A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times.
A bag contains 7 red, 5 white and 8 black balls. If four balls are drawn one by one with replacement, what is the probability that none is white ?
Find the probability distribution of the number of doublets in 4 throws of a pair of dice.
Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the mean and variance of number of red cards.
A card is drawn and replaced in an ordinary pack of 52 cards. How many times must a card be drawn so that (i) there is at least an even chance of drawing a heart (ii) the probability of drawing a heart is greater than 3/4?
An unbiased coin is tossed 8 times. Find, by using binomial distribution, the probability of getting at least 6 heads.
Suppose X has a binomial distribution with n = 6 and \[p = \frac{1}{2} .\] Show that X = 3 is the most likely outcome.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \[\frac{1}{100} .\] What is the probability that he will win a prize at least twice.
The probability of a shooter hitting a target is \[\frac{3}{4} .\] How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99?
How many times must a man toss a fair coin so that the probability of having at least one head is more than 90% ?
The probability of a man hitting a target is 0.25. He shoots 7 times. What is the probability of his hitting at least twice?
A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.
The mean and variance of a binomial variate with parameters n and p are 16 and 8, respectively. Find P (X = 0), P (X = 1) and P (X ≥ 2).
If in a binomial distribution mean is 5 and variance is 4, write the number of trials.
The mean of a binomial distribution is 10 and its standard deviation is 2; write the value of q.
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.
A fair die is thrown twenty times. The probability that on the tenth throw the fourth six appears is
A five-digit number is written down at random. The probability that the number is divisible by 5, and no two consecutive digits are identical, is
Mark the correct alternative in the following question:
The probability of guessing correctly at least 8 out of 10 answers of a true false type examination is
Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that none is a spade ?
A bag contains 7 red, 5 white and 8 black balls. If four balls are drawn one by one with replacement, what is the probability that all are white ?
For Bernoulli Distribution, state formula for E(X) and V(X).
One of the condition of Bernoulli trials is that the trials are independent of each other.
The mean, median and mode for binomial distribution will be equal when
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is:-
A student is given a quiz with 10 true or false questions and he answers by sheer guessing. If X is the number of questions answered correctly write the p.m.f. of X. If the student passes the quiz by getting 7 or more correct answers what is the probability that the student passes the quiz?
If the sum of mean and variance of a binomial distribution is `25/9` for 5 trials, find p.
The mean and variance of binomial distribution are 4 and 2 respectively. Find the probability of two successes.
