Solve the Following Question.(4 Marks)

Question 14 Marks

A random variable $X$ has the following probability distribution.

X	$1$	$2$	$3$	$4$	$5$	$6$	$7$
P(x)	$K$	$2K$	$2K$	$3K$	$K^2$	$2K^2$	$7 K^2+K$

Determine (i) $k$, (ii) $P(X < 3)$, (iii) $P(X > 6)$, (iv) $P(0 < X < 3)$.

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Question 24 Marks

Find the probability of the number of successes in two tosses of a die, where success is defined as (i) number greater than 4 (ii) six appearing in at least one toss.

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Question 34 Marks

Determine k if
$
f(x)=\left\{\begin{array}{ll}
k e^{-\theta x} & \text { for } 0 \leq x<\infty, \theta>0 \\
0 & \text { otherwise }
\end{array}\right.
$
is the p.d.f. of the r.v. $X$. Also find $P\left(X>\frac{1}{\theta}\right)$. Find $M$ if $P(0 < X < M)$=$\frac{1}{2}$

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Question 44 Marks

Let the p.m.f. of the r.v. X be

$p(x)= \begin{cases}\frac{3-x}{10} & \text { for } x=-1,0,1,2 \\ 0 & \text { otherwise }\end{cases}$

Calculate E(X) and Var(X).

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Question 54 Marks

Given that X ~ B(n, p),
(i) if n = 10 and p = 0.4, find E(X) and Var(X).
(ii) if p = 0.6 and E(X) = 6, find n and Var(X).
(iii) if n = 25, E(X) = 10, find p and Var(X).
(iv) if n = 10, E(X) = 8, find Var(X).

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Question 64 Marks

Find the probability of throwing at most 2 sixes in 6 throws of a single die.

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Question 74 Marks

The probability that a bulb produced by a factory will use fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.

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Question 84 Marks

The p.d.f. of a continuous r.v. $X$ is
$
f(x)=\left\{\begin{array}{cl}
\frac{3 x^2}{8} & \text { for } 0<x<2 \\
0 & \text { otherwise }
\end{array}\right.
$

Determine the c.d.f. of $X$ and hence find (i) $P (X<1)$, (ii) $P (X<-2)$, (iii) $P (X>$ 0), (iv) $P (1< X <2)$.

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Question 94 Marks

Suppose $X$ is the waiting time $($in minutes$)$ for a bus and its $\text{p.d.f.}$ is given by
$f(x)= \begin{cases}\frac{1}{5} & \text { for } 0 \leq x \leq 5 \\ 0 & \text { otherwise }\end{cases}$
Find the probability that $(i)$ waiting time is between $1$ and $3$ minutes, $(ii)$ waiting time is more than $4$ minutes.

Answer

$\text{p.d.f.}$ of $r.v. X$ is given by
$f(x)=\frac{1}{5}$ for $0 \leq x \leq 5$
This is a constant function.
$(i)$ Probability that waiting time $X$ is between $1$ and $3$ minutes
i.e. $ P (1< X <3) =\int_1^3 f(x) d x$
$ =\int_1^3 \frac{1}{5} d x$
$ =\frac{1}{5}[x]_1^3 $
$=\frac{1}{5}[3-1]$
$=\frac{2}{5}$
$=0.4 (ii) $ Probability that waiting time $X$ is more than $4$ minutes
$ \text { i.e. } P ( X >4) =\int_4^5 f(x) d x$
$ =\int_4^5 \frac{1}{5} d x$
$ =\frac{1}{5}[x]_4^5$
$ =\frac{1}{5}(5-4) $
$ =\frac{1}{5}$
$=0.2 $

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Question 104 Marks

$f(x)=\left\{\begin{array}{ll}k x(1-x) & \text { for } 0<x<1 \\ 0 & \text { otherwise }\end{array}\right.$
Also find (a) P $\left[\frac{1}{4}<X<\frac{1}{2}\right]$, (b) $P\left[X<\frac{1}{2}\right]$

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Question 114 Marks

A coin is biased so that the head is 3 times as likely to occur as the tail. Find the probability distribution of a number of tails in two tosses.

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Question 124 Marks

A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. If X denotes the age of a randomly selected student, find the probability distribution of X. Find the mean and variance of X.

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Question 134 Marks

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance of X.

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Maths (commerce) Solve the Following Question.(4 Marks)

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