• A random variable X is a function defined on a sample space S into the real numbersℝ such that the inverse image of points or subset or interval of ℝ is an event in S, for which probability is assigned.
• A random variable X is defined on a sample space S into the real numbers ℝ is called discrete random variable if the range of X is countable, that is, it can assume only a finite or countably infinite number of values, where every value in the set S has positive probability with total one.
• If X is a discrete random variable with discrete values x1, x2, x3,... xn..., then the function denoted by f(.) or p(.) and defined by f(xk) = P(X = xk) for k = 1,2,3,...n,... is called the probability mass function of X
• The function f(x) is a probability mass function if
(i) f(xk) ≥ 0 for k = 1,2,3,...n,... and (ii) ∑k f (xk ) = 1
• The cumulative distribution function F(x) of a discrete random variable X, taking the values x1, x2, x3,... such that x1 < x2 < x3 < … with probability mass function f(xi) is
F (x) = P(X ≤ x) = ∑xi≤x f (xi ), x ∈ ℝ
• Suppose X is a discrete random variable taking the values x1, x2, x3,... such that x1 < x2 < x3,... and F(xi) is the distribution function. Then the probability mass function f(xi) is given by f(xi) = F(xi) – F(xi–1), i = 1,2,3, ...
• Let S be a sample space and let a random variable X : S → ℝ that takes any value in a set I of ℝ. Then X is called a continuous random variable if P(X = x) = 0 for every x in I.
• A non-negative real valued function f(x) is said to be a probability density function if, for each possible outcome x, x ∈ [a,b] of a continuous random variable X having the property.
P(a ≤ x ≤ b) = b∫a f (x) dx
• Suppose F(x) is the distribution function of a continuous random variable X. Then the
probability density function f(x) is given by
f(x) = dF(x) / dx = F′(x) , whenever derivative exists.
• Suppose X is a random variable with probability mass or density function f(x) The expected value or mean or mathematical expectation of X, denoted by E(x) or μ is
• The variance of the random variable X denoted by V(X) or σ2 (or σx2) is
V(X) = E(X – E(X))2 = E(X – μ)2
(i) E(aX + b) = aE(X) + b,
where a and b are constants
Corollary 1: E(aX) = aE(X)
( when b = 0)
Corollary 2: E(b) = b
(when a = 0)
(ii) Var(X) = E(X2) – (E(X))2
(iii) Var(aX + b) = a2Var(X) where a and b are constants
Corollary 3: V(aX ) = a2V(X) (when b = 0)
Corollary 4: V(b) = 0 (when a = 0)
• Let X be a random variable associated with a Bernoulli trial by defining it as X (success) = 1 and X (failure) = 0, such that
• X is called a Bernoulli random variable and f(x) is called the Bernoulli distribution.
• If X is a Bernoulli’s random variable which follows Bernoulli distribution with parameter p, the mean μ and variance σ 2 are
μ=p and σ2 = pq
• A discrete random variable X is called binomial random variable, if X is the number of successes in n-repeated trials such that
(i) The n- repeated trials are independent and n is finite
(ii) Each trial results only two possible outcomes, labelled as ‘success’ or ‘failure’
(iii) The probability of a success in each trial, denoted as p, remains constant
• The binomial random variable X equals the number of successes with probability p for a success and q = 1 – p for a failure in n-independent trials, has a binomial distribution denoted by X ~ B(n, p). The probability mass function of X is f (x) = px (1 − p)n− x , x = 0,1, 2,..., n.
• If X is a binomial random variable which follows binomial distribution with parameters p and n, the mean μ and variance σ2 are μ = np and σ2 = np(1 – p).