Introduction Random Experiment: If in each trial of an random experiment conducted under identical conditions, the outcome is not unique, but may be any of the possible outcomes, then such an experiment is called a random experiment.
Random Variables: A variables whose values determined by the outcome of a random experiment is called a random variable.

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A random variable may be classified as either discrete or continuous depending upon the specific numerical values it can have.
A random variable that may only take on a finite or countable number of different values is referred to as a “ discrete random variable”.
Example : number of automobile sold per month, number of customers that enter a bank during one hour of operation.
“A continuous random variable” is one that can assume the infinitely large number of values corresponding to the point on the line interval.
Examples: Weight, time, temperature, life of an electric bulb.

Probability Distribution of Random Variables…… :

Probability Distribution of Random Variables…… Any representation of every possible value of random variable and the associated probabilities is called a probability distribution. Since the probabilities are assigned to every possible value of random variable, the sum of all the corresponding probabilities should be equal to one.
Probability distribution of a random variable are closely related to frequency distribution.

Probability Mass Function(p.m.f) :

Probability Mass Function(p.m.f) Probability distribution of discrete random variables are called p.m.f.
It has the following properties
If X is a discrete random variable with values x1, x2,……xn, then
P(xi) >= 0, for i = 1,2,3,…n
P(x1)+P(X2)+…….= 1

Question..1 :

Question..1 A box contains 12 bulbs of which 3 are defective. If 3 bulbs are drawn from the box at random, find the probability distribution of the number of defective bulbs drawn.
No bulbs defective
Two bulbs defective.
All three bulbs defective.

Question.2 :

Question.2 A random variable X follows the following probability distribution
Find the value of k and then evaluate P(X<6) and P(X>=6)

Expected value of a random variable :

Expected value of a random variable The expected value of a discrete random variable is a weighted average of all the possible values of the random variable, were weights are the probability associated with the values.
Mathematically:
E(X)=p1x1 +p2x2 +….pnxn = ∑ pixi
The expected value of X is also called the mean of X or the expectation of X.

Question 3 :

Question 3 From an urn containing three white balls and two black balls, a player draws two balls at random, with replacement. He gets Rs. 100 for every white ball he draws and losses Rs.50 for every black ball. Assuming he s interested in maximizing his expected profits, should he participate in such a game ?

Variance of a random variable :

Variance of a random variable When experiments are repeated over an over and for long – term, only average (mean) does not supply sufficient information and how the outcomes disperse from one performance to another. For this we need a measure of variability. The most important measure of variability is the variance.
The variance of a random variable is denoted by Var(X) or σx2. .
Mathematically, (when all values are known)
Var(X) = p1(x1-u)2+p2(x2-u)2+…+pk(xk-u)2
Where u is the expected value of X.

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Alternatively, the following simplified and more convenient computational formula for the variance is :
Var(X) = ∑x2P(x)- {E(X)}2
The standard deviation of a random variable, is the square root of the variance.

Question 4 :

Question 4 A company introduces a new product in the market and expects to make a profit of 2.5 crores during the first year if the demand is ‘good’, Rs 1.5 crores if the demand is ‘moderate’ and a loss of Rs. 1 crore if the demand is ‘poor’. Market research studies indicate that the probability for the demand to be good and moderate are 0.2 and 0.5 respectively. Find the company’s expected profit and the standard deviation.

Question 5 :

Question 5 Daily demand for transistors is having the following probability distribution..
Determine the expected daily demand of transistors. Also obtain the variance of the demand.

Probability Density Function(p.d.f) :

Probability Density Function(p.d.f) To assign the probability measure to a continuous random variable , a continuous probability distribution called p.d.f is used.
Def: A p.d.f. is any function f(x) of a continuous random variable X, where
f(x)>=0 for all x, and
The total area under the curve f(x) = 1

Question- 6 :

Question- 6 The time, X (hours), required for scheduled routine maintenance for each airplane in the fleet of International Airlines follows the probability density function:
F(x) = 3/500(10x-x2), for 0 =< x<= 10
Verify that ∫ f(x)dx=1
Find the probability that an airplane , selected at random from the fleet, requires between one and three hours for scheduled routine maintenance.
Determine expected value and variance.

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