Find the mean var

Mean of any probability distribution is given by Mean = ∑x_ip_i

Standard Deviation is given by SD = √ Variance where variance is given by:

Variance = ∑ x_i²p_i – (∑x_ip_i)²

∴ first we need to find the products i.e. p_ix_i and p_ix_i² and add them to get mean and apply the above formula to get the variance.

∴ p₁x₁ = ap and p₂x₂ = bq Similarly p₁x₁²= a²p and p₂x₂²= b²q

∴ Mean = ap + bq

Variance = a²p + b²q – (ap + bq)²

[∵ p+q = 1 ……given]

=

∴ SD = √{pq(a-b)² } = |a-b|√pq