A unique numerical value is associated with every outcome of an experiment. The value varies with every trial of the experiment. A random experiment is one whose outcome is unknown. The experiment is usually repeated under identical conditions. In an example of a coin being tossed, the outcome is a head or a tail. Every time the coin is tossed, the outcome is either of the two. The outcome is not known in prior and is unpredictable. A random variable with values being finite is a discrete random variable. A variable, which can take any value between the given limits is a continuous random variable.
How to Define Mean And Variance Of Random Variables?
The expectation of a random variable is a weighted average of all the values of a random variable. Here, the weights correspond to their respective probabilities. The formula for finding the mean of a random variable is as follows:
E (X) = μ = Σi xi pi, where i = 1, 2, …, n
E (X) = x1p1 + x2p2 + … + xnpn, where p refers to the probabilities
Variance gives the distance of a random variable from the mean. The smaller the variance, the random variable is closer to the mean.
The formula for finding the variance of a random variable is
σx2 = Var (X) = ∑i (xi − μ)2 p(xi) = E (X − μ)2 or, Var(X) = E(X2) − [E(X)]2.
E(X2) = ∑i xi2 p(xi), and [E(X)]2 = [∑i xi p(xi)]2 = μ2.
Mean And Variance Of Random Variables Examples
Examples on expectation of random variables
1] Let X represent the outcome of a roll of a fair six-sided die. More specifically, X will be the number of pips showing on the top face of the die after the toss. The possible values for X are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of 1/6. The expectation of X is
If one rolls the die n times and computes the average (arithmetic mean) of the results, then as n grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers.
2] The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable X represents the (monetary) outcome of a $1 bet on a single number (“straight up” bet). If the bet wins (which happens with probability 1/38 in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be
That is, the bet of $1 stands to lose
Mean And Variance Of Random Variables Problems
Example 1: If X and Y be the random variables with the distributions,
Then find the expected value of X and Y.
E (x) = ∑ Xi Pi
= 2 × (0.2) + 3 × 0.2 + 6 × 0.5 + 10 × 0.1
= 0.4 + 0.6 + 3.0 + 1.0 = 5
E (Y) = ∑ Yi Pi
= (0.8) × (0.2) + (0.2) (0.3) + 0 × 0.1 + 3 × 0.3 + 7 × 0.1
= 0.16 + 0.06 + 0 + 0.9 + 0.7 = 0.38
Example 2: There are 4 numbered cards beginning with 1. Two cards need to be drawn without replacement. If x denotes the sum of the numbers on the 2 cards drawn, find the mean and variance of x.
The possible sums are 3, 4, 5, 6, 7
x 3 4 5 6 7
P(x) 2/12 2/12 4/12 2/12 2/12
x * P(x) 3/6 4/6 5/3 6/6 7/6
x2 * P(x) 9/6 16/6 25/3 36/6 49/6
Mean = ∑ x * P(x) = 30/6 = 5
Variance = ∑ x2 * P(x) – [∑ x * P(x)]2 = [160/6] – 25 = 5/3
Example 3: Consider a meeting where a proposal is announced. Out of them, 70% of the members are in favour and 30% are against it. If a member is selected at random, taking X = 0 if he opposed, and X = 1 if he is in favour, find the mean and variance of X.
Here, the distribution of random variable X is:
P (X = 0) = 30/100 = 0.3 and P(X = 1) = 70/100 = 0.7
Mean = E (X) = 0 * 0.3 + 1 * 0.7 = 0.7
E (X2) = 02 * 0.3 + 12 * 0.7 = 0.7
Var (X) = E (X2) – [E (X)]2
= 0.7 – [0.7]2
= 0.7 – 0.49
Example 4: A pair of dice is thrown and let X be the random variable which represents the sum of the numbers that appear on the two dice. Find the mean of X.
The S be the sample space when a pair of dice is thrown.
; n(S) = 36
The random variable X i.e. the sum of the numbers on the two dice takes the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.
|X or xi||2||3||4||5||6||7||8||9||10||11||12|
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