5 Mark Each

Question 15 Marks

The mean of a binomial distribution is three times its variance. If the sum of its mean and variance is $8,$ find the parameters of the distribution.

Answer

$n=9, p=\frac{2}{3}$

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Question 25 Marks

A die is made in such a way that the probability of getting an even number on it is double
the probability of getting an odd number. If this die is thrown $4$ times, find the
probability of $(1)$ not getting an odd number even once $(2)$ getting less than three even
numbers.

Answer

$(1)\ \frac{16}{81}\ (2)\ \frac{11}{27}$

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Question 35 Marks

The probability that a patient gets reaction due to a medicine used for a certain disease is
$\frac{1}{10}$. If this medicine is administered to $5$ patients, find the probability that
$(1)$ two of them get reaction
$(2)$ four of them do not get reaction
$(3)$ at the most one of them gets reaction

Answer

$(1)\ 0.0729\ (2)\ 0.3281\ (3)\ 0.9186$

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Question 45 Marks

The probability distribution of a random variable $X$ is given below. $ \begin{aligned} p(x) &=k(x+3), & x=-2,-1 \\ &=k(x+2), & x=0,1,2 \\ &=k(x-1), & x=3 \end{aligned} $ Find $(1)$ the value of $k\ (2)\ p(x>0)\ (3)\ p(-1 \leq x<2)$.

Answer

$(1) \frac{1}{14} (2) \frac{9}{14} (3) \frac{1}{2}$

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Question 55 Marks

The probability mass function of a random variable $X$ is as follows: $p(x)=c x^2 , x=0,1,2,3,4,5$
Find the value of constant $c$. Also find the mean and standard deviation of variable $X$.

Answer

$c=\frac{1}{55}$, Mean $=4.09$, Standard deviation $=1.03$

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Statistics 5 Mark Each

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