A random variable X has the following probability function

(i) Find a, Evaluate
(ii) P(X < 3),
(iii) P(X > 2) and
(iv) P(2 < X \(\leq\) 5).

Construct the distribution function for the discrete random variable X whose probability distribution is given below. Also draw a graph of p(x) and F(x).

A continuous random variable X has p.d.f
f(x) = 5x⁴, 0\(\le\)x\(\le\)1 
Find a₁ and a₂ such that
i) P[X\(\le\)a₁] = P[X>a₁]   
ii) P[X>a₂] = 0.05

The amount of bread (in hundreds of pounds) x that a certain bakery is able to sell in a day is found to be a numerical valued random phenomenon, with a probability function specified by the probability density function f(x) is given  by
\(f(x)=\left\{\begin{array}{l} Ax,for \ 0≤x10 \\ A(20−x),for \ 10 ≤x< 20 \\ 0,\quad \quad \quad otherwise \end{array}\right.\)
(a) Find the value of A.
(b) What is the probability that the number of pounds of bread that will be sold tomorrow is
(i) More than 10 pounds,
(ii) Less than 10 pounds, and
(iii) Between 5 and 15 pounds?

A continuous random variable X has the following probability function

(i) Find k
(ii) Ealuate p(x<6), p(x\(\ge \)6) and p(0)
(iii) If P(X\(\le\)x).\(\frac{1}{2}\), then find the minimum value of x.

The distribution of a continuous random variable X in range (–3, 3) is given by p.d.f.
\(f(x)=\left\{\begin{array}{l} \frac{1}{16}(3+x)^{2},-3 \leq x \leq-1 \\ \frac{1}{16}\left(6-2 x^{2}\right),-1 \leq x \leq 1 \\ \frac{1}{16}(3-x)^{2}, 1 \leq x \leq 3 \end{array}\right.\)
Verify that the area under the curve is unity.

A continuous random variable X has the following distribution function:
\(f(x)=\left\{\begin{array}{l} 0 , \text{if} \ x \leq1 \\ k(x-1)^4, \text{if} \ 1< x \leq 3 \\ 1, \text{if} \ x > 3 \end{array}\right.\)
Find (i) k and (ii) the probability density function.

The length of time (in minutes) that a certain person speaks on the telephone is found to be random phenomenon, with a probability function specified by the probability density function f(x) as \( f(x)\begin{cases} { Ae }^{ -x/5 },\quad \text{for}\quad x\ge 0 \\ 0 \quad ,\quad \text{otherwise }\end{cases}\)
(a) Find the value of A that makes fix) a p.d.f,
(b) What is the probability that the number of minutes that person will talk over the phone is
(i) more than 10 minutes
(ii) less than 5 minutes and
(iii) between 5 and 10 minutes.

Suppose that the time in minutes that a person has to wait at a certain station for a train is found to be a random phenomenon with a probability function specified by the distribution function\(F(x)\begin{cases} 0,\quad ​​​​\text{for}\quad x<0 \\ \frac { 1 }{ 2 } ,\quad ​​​​\text{for}\quad 0\le x<1 \\ 0,\quad ​​​​\text{for}\quad 1\le x<2\quad \\ \frac { 1 }{ 4 } ,\quad ​​​​\text{for}\quad 2\le x<4 \\ 0,\quad ​​​​\text{for}\quad x\ge 4 \end{cases}\)
(a) Is the distribution function continuous? If so, give its probability density function?
(b) What is the probability that a person will have to wait
(i) more than 3 minutes,
(ii) less than 3 minutes and
(iii) between 1 and 3 minutes?

Determine the mean and variance of the random variable X having the following probability distribution.

Determine the mean and variance of a discrete random variable, given its distribution as follows.

Suppose the life in hours of a radio tube has the probability density function
\(f(x)=\left\{\begin{array}{l} e^{-\frac{x}{100}}, \text { when } x \geq 100 \\ 0, \quad \text { when } x<100 \end{array}\right.\)
Find the mean of the life of a radio tube.

The probability density function of a random variable X is f(x) = ke^-|x|, -∞ < x < ∞ Find the value of k and also find mean and variance for the random variable.

The probability function of a random variable X is given by
\(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
Evaluate the following probabilities.
P(\(\le\))0

Let X be a random variable with cumulative distribution function
\(F(x)=\left\{\begin{array}{l} 0, \text { if } x<0 \\ \frac{x}{8}, \text { if } 0 \leq x<1 \\ \frac{1}{4}+\frac{x}{8}, \text { if } 1 \leq x<2 \\ \frac{3}{4}+\frac{x}{12}, \text { if } 2 \leq x<3 \\ 1, \text { for } 3 \leq x \end{array}\right.\)
(a) Compute: (i) P(1\(\le\)X\(\le\)2) and 
(ii) P(X=3)
(b) Is X a discrete random variable? Justify your answer.

The probability density function of a continuous random variable X is
\(f(x)=\left\{\begin{array}{l} a+b x^{2}, 0 \leq x \leq 1 \\ 0, \text { otherwise } \end{array}\right.\)
where a and b are some constants. Find
(i) a and b if E(X)\(\frac{3}{5}\)
(ii) Var(X).

The probability function of a random variable X is given by
\(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
Evaluate the following probabilities.
P(X<0)

The probability function of a random variable X is given by
\(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
Evaluate the following probabilities.
P(|X|\(\le\)2)

The probability function of a random variable X is given by
\(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
Evaluate the following probabilities.
 P(0\(\le\)X\(\le\)10)