Expectation & Variance of a Discrete Random Variable

We can get calculate the mean from the results of an experiment. An equivalent number from a probability distribution is called the expectation or expected value. Expectation 𝐸(𝑋) Multiply each value of 𝑋 by its corresponding probability. Add these together A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05

We can get calculate the mean from the results of an experiment. An equivalent number from a probability distribution is called the expectation or expected value. Expectation 𝐸(𝑋) Multiply each value of 𝑋 by its corresponding probability. Add these together A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05

We can get calculate the mean from the results of an experiment. An equivalent number from a probability distribution is called the expectation or expected value. Expectation 𝐸(𝑋) Multiply each value of 𝑋 by its corresponding probability. Add these together A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05

We can get calculate the mean from the results of an experiment. An equivalent number from a probability distribution is called the expectation or expected value. Expectation 𝐸(𝑋) Multiply each value of 𝑋 by its corresponding probability. Add these together A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05

We can get calculate the mean from the results of an experiment. An equivalent number from a probability distribution is called the expectation or expected value. Expectation 𝐸(𝑋) Multiply each value of 𝑋 by its corresponding probability. Add these together A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05

We can get calculate the mean from the results of an experiment. An equivalent number from a probability distribution is called the expectation or expected value. Expectation 𝐸(𝑋) Multiply each value of 𝑋 by its corresponding probability. Add these together A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05

We can get calculate the mean from the results of an experiment. An equivalent number from a probability distribution is called the expectation or expected value. Expectation 𝐸(𝑋) Multiply each value of 𝑋 by its corresponding probability. Add these together A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05 =−𝟎.𝟐

You may see the formula 𝐸 𝑋 =∑𝑥𝑃(𝑋=𝑥) A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05 =−𝟎.𝟐 You may see the formula 𝐸 𝑋 =∑𝑥𝑃(𝑋=𝑥) Sum of

You may see the formula 𝐸 𝑋 =∑𝑥𝑃(𝑋=𝑥) A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05 =−𝟎.𝟐 You may see the formula 𝐸 𝑋 =∑𝑥𝑃(𝑋=𝑥) Sum of each 𝑥 times

You may see the formula 𝐸 𝑋 =∑𝑥𝑃(𝑋=𝑥) A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05 =−𝟎.𝟐 You may see the formula 𝐸 𝑋 =∑𝑥𝑃(𝑋=𝑥) Sum of each 𝑥 times Probability of 𝑥

Variance You may see the formula 𝐸 𝑋 =∑𝑥𝑃(𝑋=𝑥) A d.r.v. 𝑋 has a probability distribution as shown. Find 𝐸(𝑋) 𝑥 −2 −1 1 2 𝑃(𝑋=𝑥) 0.3 0.1 0.15 0.4 0.05 𝐸 𝑋 = −2 ×0.3+ −1 ×0.1+0×0.15+1×0.4+2×0.05 =−𝟎.𝟐 You may see the formula 𝐸 𝑋 =∑𝑥𝑃(𝑋=𝑥) Sum of each 𝑥 times Probability of 𝑥 Variance

How did we find variance of a set of data previously in S1? Variance of discrete random variables How did we find variance of a set of data previously in S1?

Our earlier slide Variance and Standard Deviation These are measures of spread around the mean. They use all data values in their calculations. There are two formulas for finding the Variance (𝜎 2 ) – both always give the same result. 𝜎 2 = Σ 𝑥− 𝑥 2 𝑛 𝜎 2 = Σ 𝑥 2 𝑛 − Σ𝑥 𝑛 2 “The mean of the squares minus the square of the mean” You need to memorise both formulas. In almost every situation the second formula is easier to use.

Variance of discrete random variables We do essentially the same thing, though in S1 we use 𝑉𝑎𝑟 for a d.r.v 𝑉𝑎𝑟(𝑋)=𝐸 𝑋 2 − 𝐸 𝑋 2 The following will show how to calculate 𝐸 𝑋 2 as part of the calculation of 𝑉𝑎𝑟 How did we find variance of a set of data previously in S1? The mean of the squares minus the square of the mean 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋)

Variance of discrete random variables 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋) We need to include an 𝑥 2 row 𝑥 1 2 3 4 𝒙 𝟐 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1

Variance of discrete random variables 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋) We need to include an 𝑥 2 row 𝑥 1 2 3 4 𝑥 2 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1

Variance of discrete random variables 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋) We need to include an 𝑥 2 row 𝑥 1 2 3 4 𝑥 2 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1

Variance of discrete random variables 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋) We need to include an 𝑥 2 row 𝑥 1 2 3 4 𝑥 2 9 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1

Variance of discrete random variables 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋) We need to include an 𝑥 2 row 𝑥 1 2 3 4 𝑥 2 9 16 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1

Variance of discrete random variables 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋) We need to include an 𝑥 2 row 𝑥 1 2 3 4 𝑥 2 9 16 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 a) 𝐸 𝑋 = 1×0.4 + 2×0.2 + 3×0.3 +(4×0.1) =2.1

For 𝐸( 𝑋 2 ) we just use the 𝑥 2 row instead Variance of discrete random variables 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋) We need to include an 𝑥 2 row 𝑥 1 2 3 4 𝑥 2 9 16 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 a) 𝐸 𝑋 = 1×0.4 + 2×0.2 + 3×0.3 +(4×0.1) =2.1 For 𝐸( 𝑋 2 ) we just use the 𝑥 2 row instead

For 𝐸( 𝑋 2 ) we just use the 𝑥 2 row instead Variance of discrete random variables 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋) We need to include an 𝑥 2 row 𝑥 1 2 3 4 𝑥 2 9 16 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 a) 𝐸 𝑋 = 1×0.4 + 2×0.2 + 3×0.3 +(4×0.1) =2.1 For 𝐸( 𝑋 2 ) we just use the 𝑥 2 row instead

For 𝐸( 𝑋 2 ) we just use the 𝑥 2 row instead Variance of discrete random variables 𝑥 1 2 3 4 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 For the given distribution, find a) 𝐸 𝑋 b) 𝐸 𝑋 2 c) 𝑉𝑎𝑟(𝑋) We need to include an 𝑥 2 row 𝑥 1 2 3 4 𝑥 2 9 16 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 a) 𝐸 𝑋 = 1×0.4 + 2×0.2 + 3×0.3 +(4×0.1) =2.1 For 𝐸( 𝑋 2 ) we just use the 𝑥 2 row instead b) 𝐸 𝑋 2 = 1×0.4 + 4×0.2 + 9×0.3 +(16×0.1) =5.5

Variance of discrete random variables 𝑥 1 2 3 4 𝑥 2 9 16 𝑃(𝑋=𝑥) 0.4 0.2 0.3 0.1 a) 𝐸 𝑋 = 1×0.4 + 2×0.2 + 3×0.3 +(4×0.1) =2.1 b) 𝐸 𝑋 2 = 1×0.4 + 4×0.2 + 9×0.3 +(16×0.1) =5.5 𝑉𝑎𝑟(𝑋)=𝐸 𝑋 2 − 𝐸 𝑋 2 Ex 8C page 162 Q1-5 (about 10 – 15 mins) Ex 8D page 164 Skip Q3 c) 𝑉𝑎𝑟 𝑋 =5.5 − 2.1 2 =1.09

Hints 8C 8D Ex 8C page 162 Q1-5 (about 10 – 15 mins) Ex 8D page 164 Q3c – simply show what each is & therefore show whether equal or not Q4 – List the sample space (all the equally possible outcomes), starting HH,… Q5 – make two equations with 𝑎 and 𝑏 in each & solve simultaneously 8D Q1a – you should be able to work it out in your head! Q1b – show working for this part Q4a – Use the sample space diagram at right (draw your own in an exam if necessary) Q5 – Draw your own sample space Q6 – List sample space Q7 – Think about the expected value, by looking at the probabilities of the higher and lower values… Remember “Write down” means no working required. Ex 8C page 162 Q1-5 (about 10 – 15 mins) Ex 8D page 164 Skip Q3