Study Notes on Random Variables

Random Variable

1. Random variables X: Let S be the sample space of an experiment. A real-valued function X:S→R is called a random variable of the experiment if, for each interval I⊆R, {s:X(s)∈I} is an event.

2. Probability function p_X(x):

(a) p_X(x)=0 if x∉ range R_X;

(b) P_X(x)≥0 if x∈R_X

(c) 

Example: If a coin is tossed three times, the number of heads obtained can be 0, 1, 2 or 3. The probabilities of each of these possibilities can be tabulated as shown:

In this example, the number of heads can only take 4 values (0, 1, 2, 3) and so the variable is discrete. The variable is said to be random if the sum of the probabilities is one.

Distribution Functions

1. Cumulative Distribution Functions (cafe): F_X(t) = P(X ≤ t), -∞<t<∞.

Example:

If a die is thrown repeatedly, let’s work out P(X ≤ t) for some values of t.

P(X ≤ 1) is the probability that the number of throws until we get a 6 is less than or equal to 1. So it is either 0 or 1.

P(X = 0) = 0 and P(X = 1) = 1/6. Hence P(X ≤ 1) = 1/6

Similarly, P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0 + 1/6 + 5/36 = 11/36

2. F_X(t) is non-decreasing; 0≤F_X(t)≤1; 

3. If c<d, then F­_X(c)≤F_X(d); P(c<X≤d) = F_X(d) – F_X(c); P(X>c) = 1-F_X(c).

4. The cdf of a discrete random variable: a step function.

Mean and Variance of Random Variables

Mean:

The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome x_i according to its probability, p_i. The common symbol for the mean (also known as the expected value of X) is μ, formally defined by

μ_x = x₁p₁ + x₂p₂ +…+ x_kp_k

= ∑x_ip_i

The mean of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.

Variance:

The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by

The standard deviation  is the square root of the variance.

Expectations of Discrete Random Variables

1. Expected value (mean value or average value or expectation) for a random variable 

2. Let g be a real-valued function. Then g(X) is a random variable with 

3. Let g₁, g₂, …, g_n be real-valued functions, and let α₁, α₂, …, α_n be real number. Then E[α₁g₁(X) + α₂g₂(X) +…+ α_ng_n(X)] = α₁E[g₁(X)] + α₂E[g₂(X)] +…+ α_nE[g_n(X)].

Variances of Discrete Random Variables

1. The variance of a random variable: the average square distance between X and its mean μ_X

2. Standard deviation: 

3. Let X be a discrete random variable, then Var[X]=0 if and only if X is a constant with probability 1.

4. Let X be a discrete random variable, then for constants a and b: Var[αX+b] = a²Var[X], σ_aX+b = |a|σ_X.

Example 1:

Which of the following holds true for any probability density function (symbols have their usual meaning):-

(A). V(x) = [E(x)]² - [E(x²)]

(B). V(x) =[E(x²)] - [E(x)]²

(C). V(x) = [E(x²)] + [E(x)]²

(D). V(x) = (E(x²) - E(x))²

Correct Answer: 2

Solution:

So,

Example 2: Let f(x) = k x (1-x), for 0<=x<=1 then k is

(A). ½ (B). 1/3

(C). 1/6 (D). 6

Correct Answer: 4

Solution:

Example 3: Let f(x) = k x (1-x), for 0<=x<=1 then mean is

(A). 1/6 (B). 1/2

(C). 3/10 (D). 1/3

Correct Answer: 2

Solution:

Example 4: Let f(x) = k x (1-x), for 0<=x<=1 then variance is

(A). 3/10 (B). 1/10

(C). 1/20 (D). 1/2

Correct Answer: 3

Solution: